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n:l 67^. /X 





HARVARD 
COLLEGE 
LIBRARY 




SCIENTIFIC ROMANCES. 

By C HowAED HiNTON, M.A. 

Cromi 8vo» cloth gilt, 6r. i or •qMumteljr, u. each. 

I. What it the Pottrth Dimentloii? ii. 

GHOSTS EXPLAINED. 

A MMVt tfwitiM Of idwiinibw oMfMM* • • • Mr* H iBlon brinfli utf 

tat ddigkMd. tP at kaat a momentanr Ikkh ia Um FouitJi Di- 

, wmi npoB lh« cy« of this faith ihert opeat a vitta of iatorMliaf 

J. . . . Hm paaqrfilet cshibils a boldnoM of spocolatioo, aad a 

of eoMciviiiff and tx|ireMinj| CT«a tta isooMtivaMt, whidi f oa m 



Mi^t liKakioa Kta a'l0Mic''-:/w7V«i/. 
2. TlM Persian KiaE* or, The Lew of the Valley, i/. 
THE MYSTERY OF PLEASURE AND PAIN. 
Atwy iiig|Mliv« aad ivcll*irriiMa i po cu l ati oB, hf tta iahoiilor of aa 




Will ancac tta autatioa of tta nador at oaca.'^ATMMidtn^pr. 

3. A Plena World, ii. 

4. A Picture of our Universe, i/. 

$• Cestinf ottt the Self, i#. 

SMCOJfP SMMiMS. 

I. On the Bdveeiion of the Imefinatien. 
a. If any Dimensions, ia 



LONDON: SWAN 80NNEN6CHEIN ft Ca 



^^ 



^ Nm Era of Thought 



7?- 



'-; ! 



CHARLES HOWARD mjITO^A, OXON. 




IPoitkoii: 
SWAN SONNENSCHEIN & CO.- 
* JP^'^RNOSTER SQUARE. 

9-- 



A NEW ERA OF THOUGHT. 



I 472. i5 




'^^^^^^ jf^^-yvoC. 



fM tuwoM rBar 




) f 








1 ,• 




1 \\ 


1 



PREFACE. 



The MSS. which formed the basis of this book were 
committed to us by the author, on hiis leaving England 
for a distant foreign appointment ' It was his wish that 
we should construct upon them a much more complete 
treatise than we have effected, and with that intention 
he asked us to make any changes or additions we thought 
desirable. But long alliance with him in this work has 
convinced us that his thought (especially that of a general 
philosophical character) loses much of its force if sub- 
jected to any extraneous touch. 

This feeling has induced us to print Part I. almost 
exactly as it came from his hands, although it would 
probably have received much rearrangement if he could 
have watched it through the press himself 

Part 11. has been written from a hurried sketch, which 
he considered very inadequate, and which we have con- 
sequently corrected and supplemented Chapter XL of 
this part has been entirely re-written by us, and has thus 
not had the advantage of his supervision. This remark 
also applies to Appendix E, which is an elaboration of 
a theorem he suggested. Appendix H, .and all the 
exercises have, in accordance with his wbh, been written 



VI 



Preface. 



solely bf us. It will be apparent to the reader that 
Appendix H is little more than a brief introduction to 
a very laige subject, which, being concerned with tes- 
saracts and solids^ is really beyond treatment in writing 
and diagrams. 

This difficulty recalls us to the one great fact, upon 
which we feel bound to insist, that the matter of this 
book nmst receive objective treatment from the reader, 
who will find it quite useless even to attempt to appre- 
hend it without actually building in squares and cubes 
all the 4acts of space which we ask him to impress on 
his consciousness. Indeed, we consider that printing, 
as a method of spreading space-knowledge^ is but a ^pis 
aller," and we would go back to that ancient and more 
fruitful method of the Greek geometers, and, while 
describing figures on the sand, or piling up pebbles in 
serieai» would communicate to others that spirit of learn- 
ing and generalisation bq^otten in our consciousness by 
continuous contact with facts^ and only by continuous 
contact with facts vitally maintained 

ALICIA BOOLE, 
H. JOHN FALK. 

N.& ModelsL — It is unquestionably a most important 
part fA die process of learning space to construct theses 
and the reader should do so^ however roughly and 
hastily. But; if Modeb are required as patterns, they 

ly be ordered from Messni Swan Sonnenschein & Ca» 



Preface. 



VII 



Paten«.ter Square. London, and will be .«PpU«i - 
L a. possible, the uncertainty a, to demand f^ 
lenotS^ingustobave a large «-ber m^e m 
advance. Much of the work can be done w.A pl«n 
t^ by using name, without colour^ ^^'^Z 
the re«ier wiU find colour. «ece.«.ry to ^-^^J^ 
^p «.d retain the complex .erie. of obser^Uons. 
CoJured model, can easily be mad. ^3r coving ^n- 
dergarten cube, with white paper «<» J^rj^^"* 
tirwater^olour. and. if permanence be defied, dip- 
ping them in «xe and copal vamirfu 



TABLE OF CONTENTS. 



PART I. 

. IlfTRODUCnON • 1-7 

CHAPTER h 
Scepcidfin and Sdenoe. Beginning of Knowledgt • 8-13 

CHAPTER IL 

Apprehension of Nature. InteUigenceb . Study of Anange- 

ment or Shape t4-ao 

CHAPTER HI. 
The'EIements of KnowMge • • .^ . . . 31-33 



CHAPTER IV. 
Theoiy and Practice , 



34-38 



CHAPTER V. 
Knowledge: Self-Elements « 



^9-34 



CHAPTER VI. 

Function of Mind. Space against Metaphysics. Self* 

Limitation and its Test A Plane World . *• . 3S-46 

Ix 



Sdf 



ConUnts, 

CHAPTER VII. 
in our CoDidoiitiieM . 



47-50 



CHAPTER VIII. 
y Rdalioo of Loww to Higher Space. Tbecnyoftlieifither si-^ 

CHAPTER IX. 
AM^er View of the iEther. Material and iEtherial Bodies 61-^ 

CHAPTER X. 

Higher Spaoe and Higher Being. Perception and !»• 

•piintXMi 67-<4 

CHAPTER XL 
Space the Scientific Basil of Altruism and Religion • . 85-99 



PART 11. • 

CHAPTER I. 

Th ite sp a ce. Genesis of n Cube. Appearances of a Cube 

to n Plane-being 101*113 

CHAPTER II. 
Farther Appearances of a Cube ton Plane-being . 113-117 

CHAPTER IIL 

Fow-qinoe. Genesis of n Tessaract ; its Repntsentation 

hi Thiea-spnce 118-129 

■^ CilAPTER IV. 

TcHanct nwving through Three-space, liodelsofthe 

. . . . ... . . . 130-134 



Contents. 



XI 



156-166 



CHAPTER V. 
R,p«.entationofTliree-spacebyNa«es.ndlnaPtoe .3$-«4S 

CHAPTER VI. 
TheMeansbywhIchaPlane-beingwonldAcqulniaCon. ^^^^^^ 

ception of oor Figures 

CHAPTER VII. 
Fonr^paceilU Representation in Three-space . 

CHAPTER Vlll. 
Rep^senutlonofFonr-spacebyName. Study of Tessaract. «67-.76 

CHAPTER IX. 
Further Study of Tcssaracts . . • • 

CHAPTER X. 
Cyclical Projections 

CHAPTER XL 
A Tessaractic Figure and iu Projections 



. i77->79 



i8eKiS3 



. 184-194 



APPENDICES. 

197 

A, 100 Namei used for Plane Spwe . • ' ^^ 

B. ai6 Names used for CuWc Space . • ^, 
C as6 Name, used for TessMW^cS^ • 

D. List of Colours, Names, and SymboU . ^^^ 

E. A Theorem in Four-space . •. • * 20«-307 
Y. Exercises on Shapes of Three D«2«^ • ; ^^ 
G. Exerdse. on Shapes of Four Dm>ensK«. . • ^^^ 

^«cr(ModeUi-«a)wUh.Colours and Names . ai9-a4i 



INTRODUCTORY NOTE TO PART I. 



At the completion of a work, or at the completioii of the fint part 
of a work, the feelings are necessarily very different from those 
with which the work was begun ; and the meaning and vaiae of the 
work itself bear a very different appearance. It will therefore be 
the nmplest and shortest plan, if I tell the reader briefly what the 
work is to which these pages are a guide, and what I consider to 
• be its value when done. 

The task was to obtain a sense of the properties of higher space, 
or space of four dimensions, in the same way as that by which we 
reach a sense of our ordinary three-dimensional space. I now prefer 
to call the task that of obtaining a familiarity with higher matter, 
which shall be as intuitive to the mind as that of ordinary matter 
has become. The expression ** higher matter* is preferable to 
** higher space," because it is a somewhat hasty proceeding to split 
this concrete matter, which we touch and feel, into the abstractions 
of extension and impenetrability. . It seems to me that I cannot 
think of space without matter, and therefore, as no necessity com- 
pels me to such a course, I do not split up the concrete object into 
subtleties, but I simply ask : '* What is that which is to a cube or 
block or shape of any kind as the cube is to a square ?* 

In entering upon this inquiry we find the task is twofold. 
Firstly, there is the theoretical part, which is easy, vis, to set 
deariy before us the relktive conditions which' wouM obtain if 
there were a matter physically higher than this matter of oursi^aad 

«s 



J 



XIV 



ifuroauaary /voie to Fart A 



Introauctafy Jsote to i^an /. 



XV 



to dMNM the best means of i lil^eretj ^oor minde from the limiu- 
tiont imposed on it bjr the pArticqter conditions under which we 
■le pbced The second part of the task is somewhat bborious, 
and consists of a constant presentation to the senses of those ap- 
peaiaaces which portions of higher matter would present, and of 
a conttnoal dwdlfaig on them, until the higher matter becomes 



•/ 



The readermust undertake this task, if he accepu it at all, as an 
experiment Those of us who have done it, are satisfied that there 
is that in the resulu of the experiment which make it weU worthy 
o£a trial 

And in a few words I may sute (he general bearings of this 
work, lor every branch of work has its general bearings. It is an 
attempt, hi the most elemenUry and simple domain, to pass from* 
the kmer to the higher. In pursuing it the mind passes from one 
kind of intuition to a higher one, and with that tiansition the 
boriion of thought is altered. It becomes clear that there is a 
physical existence transcending the ordinary physical existence ; 
and one becomes faidined to think that the right direction to look 
is, not away from matter to spiritual existences, but towards the 
discovery of conceptions of higher matter, and thereby of those 
material existences whose definite relations to us are apprehended 
as spiritual intuitkms. Thus, ^'material'' woukl simply mean 
''grasped by the intellect, become known and fiuniliar.* Our ap- 
prehcnskm of anything which is not expressed in terms of matter, 
is v^gue and indefinite. To realireand live with that which we 
vaguely discern, we need to apply the intuition of higher matter to 
tfie world araond us. And this seems to me the great inducement 
totUsstudy. Let us form our intuitkm of higher space, and then 
look oat upon ihe world. 



Secoodly, fai this pragress ftom ordinary to higher matter, as a 
~ type of progress ftom lower to higher, we make the fol- 
Firstly, we beoome aware that there are 



.1 



certain limitations aflfecting our regard. Secondly, we discover by 
(^ur reason what thoM limitations are, and then force ourselves to 
go through the experience which would be oure if the limitations 
did not affect us. Thirdly, we become aware of a ci^wcity within 
us for transcending those lunitationi, and for living in the higher 
mode as we had lived in the previous one. 

We may remark that this progress from the ordinary 'to the 
higher Idnd of matter demands an absolute attention to details. It 
is only in the retention of details that such progress becomes pos- 
sible. And as, in this questkm of matter, an absolute and uncon* 
ventional examination gives us the indication of a higher, so, 
doubtless, in other questions, if we but come to fiicts without pre- 
supposition, we b^n to know that there is a higher and to dis- 
cover indications of the way whereby we can approach. That way 
lies in the Iblness of detail rather than in the generalisation. 

Biology has shown us that there is a universal order of forms 
or otiganisms, passing ftom lower to higher. Therein we find an 
indication that we ourselves take part in this progress. And in 
using the litUe cubes we can go through the process ourKlves, and 
learn what it is in a little instance. 

But of all the ways in which the confidence gained from this 
lesson can Ub applied, the nearest to us lies in the suggestion it 
gives,— and more than the suggestion, if inclination to think be 
counted for anything,— in the suggestion of that which is higher " 
than ourselves. We, as individuals, are not the limit and end*all, 
but there is a higher being than ours. What our relation to it is, 
we cannot tell, for that is unlike our reUtion to anything we know. J 
But, perhaps all that happens to us is, could we but grasp it, our 
rehoion to it. 

At any mat^ the discovery of it is the great object beside which 
all else is as secondary as the routine of mere existence Is to 
companionship. And the method of discovery is full knowledge of 
eadi other. Thereby Is the hi^ier being to be Mown. Inasmuch 



' i 



xv» IntrodMCtory Note to Part I: 

■*'™^ "^^I.^ -r ta.«.rtod« of the toett «f •«* •»»«^* f 



rl 



A NEW ERA OF THOUGHT. 



PART I. 
INTRODUCTION. 



^ 



ST; 



There are no new truths in this book, but it consists 
of an effort to impress upon and bring home to the 
mind some of the more modem developments of thought 
A few sentences 6r Kant, a few leading ideas of Gauss 
and Lobatslchewski fornwthe material out of which it 
is built up. 

It may be thought to be unduly long ; but it must 
be remembered that in these times there is a twofold 
process going on— one of discovery aboui external 
nature, one of education, by which our minds are 
brought into harmony with that which we know. In 
certain respects* we find ourselves brought on by the 
general current of ideas— we feel that matter is permanent 
and cannot be annihilated, and it is almost an axiom 
in our minds that energy is persistent, and all its trans- 
formations remains the same in amount But there are 
other directions in which there is need of definite train- * 
|ing if we are to enter into the thoughts of the time. 

And it seems to me that a return to Kant, the ^^^^^^'^l/^iji- 
i modem philosophy, is the first condition. Now of ^^^^^ 
ant's enormous work only a small part Is treated here, 
^ut with the difference that should be found between the 
wk of a master and that of a follower. Kant's state- 



I 



2 A New Era of T/tought. 

mcntB are taken as leading ideas, suggesting a field of 
work; aiid it is in detail and manipulation merely that 
there is an opportunity for workmanship. 

Of Kant's* work it is only his doctrine of space which 
is here experiinented upon. With Kant the perception 
of thiiq;s as being in space is not treated as it seems so 
obvious to da We should naturally say that there is 
space; and there are things in it From a comparison 
of those properties which are common to all things we 
obtain the properties of space But Kant says that 
diis property of being in space is not so much a quality 
of any definable objects, as the means by which we ' 
obtain an apprehension of definable ob]ects-^it is the 
condition of our mental work. 

Now as Kanfs doctrine is usually commented on, the 
negative side is brought into prominence, the positive 
side is neglected. It is generally said that the mind 
cannot pticeive things in themselves, but can only 
apprehend them subject to space conditions. And in 
this way tlie space conditions are as it were considered 
somewhat in the light of hindrances, whereby we are 
pcevented from seeing what the objects in themselves 
truly are. But if we take the statement simply as it 
ts-Hthat we apprehend by means of space — ^then it is 
equally allowable to consider our space sense as a 
positive means by which the mind grasps its experience. 

There is* in so many books in which the subject is 
treated a certain air of despondency— as if this space 
apprehension were a kind of veil which shut us off from 
nature But there is no need to adopt this feeling. 
The first postulate of this book is a full recognition of 
the fiict^ that it is by means of space that we apprehend 
what is. Space isjth eJnstnunent ofthrminA> 

And Jiere for the purposes of our work wfe can avoid 
all metaphysical discus8k>a Very often a statement 






f 

'a 



L. 



. •! 









V' 



\ 



Introduction. 3 

which seems to be very deep and abstruse and hard 
to grasp, is simply the form into which deep thinkerr 
have thrown a very simple and practical observation. 
And for the present let us look on Kant's great doctrine 
of spac^ from a practical point of view, and it comes to 
this— it is important to develop the space sense, for it is 
the means by which we think about real things. 

There is a doctrine which found much favour with 
the first followers of Kant, that also affords us a simple 
and practical rule of work. It was considered by Fichte 
that the whole external world was simply a projection 
from the ego^ and the manifold of nature was a recogni- 
tion by the spirit of itself. What this comes to as a 
practii^ rule is, that we can only understand nature in 
virtue of our own activity ; that there is no such thing 
as mere passive observation, but every act of sight and 
thought is an activity of our own. 

Now according to Kant the space sense; or the in- 
tuition of space, is the most fundamental power of the 
mind. But I do not find anywhere a systematic and 
thoroughgoing education of the space sense. In every 
practical pursuit it is needed— in some it is developed. 
In geometry it is used ; but the great reason of failure 
in education is that, instead of a systematic training 
of the space sense, it is left to be organized by ac^ 
cident and is called upon to act without having been 
formed. According to Kant and according to common 
experience it will be found that a trained thinker is one 
in whom the space sense has been well developed. 

With rqfard to the education of the space sense» I 
must ask the indulgence of the reader. It will seem 
obvious to him. that any real pursuit or. real observation 
trains the space sense, and that it is going out of the 
way to undertake any special discipline. • 

To this I would answer that, according to my own 



\ 



4 A New Era of Thought. 

•^? w ^ T ^^^ '"^"^^ ^^ ^P^ "Nations 

S^LlS^^^f' a true view of space facts a whole 

sencs of conception^ which before I had known merely 

by repute and grasped by an effort, became perfecUy 

simple and clear to me. r^ j 

Moreovw. to take one instance: in studying the 

«^ns of space we always have to do with^lJured 

objects, we always have the sense of weight ; for if the 

tting, themselves have no weight, there is always a 

direction of up and down-which implies the sense of 

A^r .? P^^^P? *^ ^^ P^^^* ^^ ^^^ to take \s 
S^if ?^ '^''^i*' *^ "P^«^ »^"^ ^«" developed 
aI w^'T-'l? ^'^"^^^ *" ^^^'^^ through the paitof 
the book which relates to it, and the phraseologf will 
scive him for the considerations which come next 

Amongst the followers of Kant, those who pursued 
SL ^l^^^i'""^. ""^ ^"^^^^ ^ ^^ ^'^ have attracted 
^T *5T''"c'?'lv ^""^ ^^ considered as his sue 
^HL Fidite, Schelhng, Hegel have developed cer- 
toin tendencies and have written remarkable books. 
|2;;^*^^tra^ and LobaNy 

I Fdf if our inttution of space is the means by which we 

'^I^^^^ *'^?"^"' ^' ^^ ^^y^ different 
kind! of intuitions of space. Who can tell what the ab- 
jolute space mtuition is ? This intuition of space musJ 

Now, after Kant had laid down his doctrine of space. 
It JIM important to investigate how much in our space 
intuition 1. due to experience-is a matter of the Jhy! 
«cal dfcumstancesof the thinking being-and how 
much is the pure act of the mind. 



Introduction. 



The only way to investigate this is the practical way, 
. and by a remarkable analysis the great geometers above 
mentioned have shown that space is not limited as or- 
dinary experience would seem to inform us, but that we 
are quite capable of conceiving different kinds of space. ^ 

Our space as we ordinarily think of it is conceived as 
limited— not in extent, but in a certain way which can 
only be realized when we think of our ways of measur« 
ing space objects. It is found that there are only three 
independent directions in which a body can be measured 
— ^it must have height, length and breadth, but it has 
no more than these dimensions. If any other measure* 
ment be taken in it, this new measurement will be found 
to be compounded of the old measurements. It is im* 
possible to find a point in the body which could not 
be arrived at by travelling in combinations of the three 
directions already taken. 

But why should space be limited to three independent 
directions ? 

Geometers have found that there is no reason why 
bodies should be thus limited. As a matter of fact all 
the bodies which we can measure are thus limited. So 
we come to this conclusion, that the space which we use 
for conceiving ordinary objects in the world ia limited 
to three dimensions. But it might be possible for there 
to be beings living in a world such that they would con- 
ceive a space of four dimensions. All that we can say 
about such a supposition is, that it is not demanded by 
our experience. It maybe that in the 'very large or 
the very minute a fourth dimension of space will have 
to be postulated to account for parts-*but with regard | 
to objects of ordinary magnitudes we are certainly not 1 
in a four dimensional world. . ' 

And this was the point at which about ten years agp 
I took up the inquiry. 



/ 






4^ 



W mw Era of Tftaught. 



^^-^^!^t^o7:j^T '"l"' ^Paceof higher 

«J^ve fo„r.din,en,lo J^^'iriT ^"* "" *« 
which we can concdv- «... I * **"* "^^Y «n 

of a body in forr!.!^:^;::;^;;^/ Can we th'ink 

«» Uie same way as we th.T^ u J ."8^ Propcrtie. 

>row this Q^i^ ,, . "* '^^ "« ^»«n'''«' 

I «»o»nienced a SWIM afJlIT- "' SSESnjnent And 
elusion one way ^S.'^thT"'"*"''^^*^^^* * «<>"• 

«^ptli:S wilfdS^S^ra laT?^" "^^ *"^"» 
•fciUed manipuktor wm fS ^ °^ "**•"*• "»« '«« 

P"<Ied on the nu^Uiw ' ,** ••«•■«• *veo^ng de. 

feirlywodd surdj^d^Tl^"* *° P"* the question 
ticalartofeduaSon *^ «^ "^urce of the pmc. 

Jay ahsol«tdydl^ra?&;^?"'-'*'°'«'*'«'»' bodies 
P^ the whJle .1^ «f^„?^.* ^« «h»nge of 

•i-- i-fectiy deiid%i:;r;o's^^^^ 



<\ 






/ 



Introductuni. 7 

that very trcith which, when first stated by Kant, seemed 
to close the mind within such fast h'mits. Our percep- 
' tion is subject to the condition of being in space. But 
space is not limited as we at first think. 
iThc next step after having formed this power of con- 
ception in ampler space, is to investigate nature and see 
what phenomena are to be explained by four*dimen* 
sienal relationsj 

But this part of the subject is hardly one for the same 
worker as the one who investigates how to think in four* 
dimensional space^ The work of building up the power 
is the work of the practical educator, the work of apply- 
ing it to nature is the work of the scientific man. And 
it is not possible to accomplish both tasks at the same 
time: Consequently the crown is still to be won. [Here 
the method is given of training the mind ; it will be an 
exhilarating moment when an investigator comes upon 
phenomena which show that external nature cannot be 
explained except by the assumption of a four-dimen*/^ 
sion space.^ 

The thought of the past ages has used the conception 
of a three-dimensional space, and by that means has 
classified many phenomena and has obtained rules for 
dealing with matters of great practical utility. The 
path which opens immediately before us in the future 
is that of applying the conception of four-dimensional 
space to the phenomena of nature, and of investigating 
what can be found out by this new means of appre- 
hension 

In fact, what has been passed through may be called 
the three-dimensional era; Gauss, and Lobatchewski 
have inaugurated the four-dimensional erd; 



1-1 



5- 



4 

I .V- 



>^fl;iU€-Ocai 



Y 



/.oiai 



CHAPTER I. 

SCEPTICISM AND SCIENCE. BEGINNING OF 
KNOWLEDGE. 

The following pag« have for thdr object to t^ 
™e ftader to apply himseir ♦» ♦!.- * 7 . *** '"*'"<^ 

•P to tiM .Und of o«" to Zl2r^ ""^ •^'« 

•--y*tor««t AuJL.'^.^^H^^iSSS 



t, 



i: 



Scepticism and Science, 9 

of knowledge--the arts, the sciencesi interesting talk, 
useful inventions — and yet I myself was profited 
nothing at all; for somehow, amidst all this activity, I 
was left alone, I could get nothing which I could knowT^ 

The dialect was foreign to me — its inner meaning 
was hidden. If I would, imitating the utterance of 
my fellows, say a few words, the effort was forced, the 
whole result was an artificiality^ and, if successful, would 
be but a plausible impostura 

The word " sceptical " has a certain unpleasant asso- 
ciation attached to it, for it has been used by so many 
people who are absolutely certain in a particular line^ 
and attack other people's convictions. But to be scep- 
tical in the real sense is a far more unpleasant state of 
mind to the sceptic than to any one of his companions. 
For to a mind that inquires into what it really does 
know, it is hardly possible to enunciate complete sen* 
tences, much less to put before it those complex ideas 
which have so large a part in true human life. 

Every word we use has so wide and fugitive a mean* 
ing, and every expression touches or rather grazes fact 
by so very minute a point, that, if we wish to start.with 
something which we do know, and thence proceed in a 
certain manner, we are forced away from the study of 
reality and driven to an artifidal system, such as logic 
or mathematics, which, starting from postulates and 
axioms, develops a body of ideal truth which rather 
comes into contact with nature than is nature. 

Scientific achievement is reserved for those who are 
content to absorb into their consciousness, by any means 
and by whatever way they come, the varied appearances 
of nature^ whence and in which by reflection they find . 
floating as it were on the sea of the unknown, ceitain 
similarities, certain resemblances and analogies, by 
means of which they collect together a body of possible 



/ 






lO 



A New Era of Thought. 



predictions and inferences ; and in nature they find 
correspondences which are actually verified Hence 
science exists^ although the conceptions in the mind 
cannot be said to have any real correspondence in 
nature. 

We form a set of conceptions in the mind, and the 
relations between these conceptions give us relations 
which we find actually vibrating in die world around 
us. But the conceptions themselves are essentially 
aitificiaL 

We have a conception of atoms ; but no one supposes 
that atoms actually exist We suppose a force vaiying 
inversely as the square of the distance ; but no one 
supposes such a mysterious thing to really be in nature. 
And when we come to the region of descriptive science, 
when we come to simple observation, we do not find 
ourselves any better provided with a real knowledge of 
nature. If, for instance, we think of a plant, we picture 
to ourselves a certain green shapes of a more or less 
definite character. This green shape enables us to 
recognise the plant we think of, and to describe it to a 
certain extent But if we inquire into our imagination 
of k; we find that our mental image very soon diverges 
from the fact If, for instance^ we cut the plant in* half, 
we find cdls and tissues of various kinds. If we examine 
our idea of the plant, it has merely an external and 
superficial resemblance to the plant itself. It is a mental 
- drawing meeting the real plant.in external appearance ; 
but the two things, the plant and our thought of it, 
oome as it were from difierent sides— they just touch 
each other as far as the colour and shape are concerned, 
but as structures and as living organisms they are as 
Wide apart as possiUe. 

Of course by observation and study the image of a 
plant which we bear in our minds may be made to re- 



Beginning of Knowledge. 



II 



I 



scmble a plant as found In the fields more and more. 
But the i^reement with nature lies in the multitude of 
points superadded on to the notion of greenness which 
we have at first— there is no natural starting-point where 
the mind meets nature, and whence th^ can travel hand 

in hand. . /. •. 

It almost seems as if, by sympathy and feehng, a 
human being was easier to know than the simplest ob- 
ject To know any object, however simple, by the reason 
and observation requires an endless process of thought 
and looking, building up the first vague impression mto 
something like in more and more respects. While, on 
the other hand, in deaHng with human beings there is 
an inward sympathy and capacity for knowing which is 
independent of, though called into play by, the obser- 
vaUon of the actions and outward appearance of the 
human being. 

But for the purpose of knowing we must leave out 
these human relationships. They are an affair of in- 
stinct and inherited unconscious experience. The mmd 
may some day rise to the level of these inherited appre- 
hensions, and be able to explain them ; but at present 
it is far more than overtasked to give an account of the 
simplest portions of matter, and is quite inadequate to 
give an account of the nature of a human being. 

Asking, then, what there was which I could know, 
I found no point of beginning. There were plenty of 
ways of accumulating observations, but none in which 
■ one could go hand in hand with nature. 

A chad is provided in the early part of its Ufe with a 
provision of food adapted for it But it seemed that our 
minds are left without a natural subsistence, for on Uie 
one hand there are arid mathematics, and on the other 
there is observation, and in observation there i% out 
ofthe great mass of constructed mental imAges, but htUe 



/ 



-R" 



Ji J})€w Era of ThouglU. 



Beginning of Knowledge. 



*^ 



which die mind can assimilate. To the worker at science 
of comfse this crude and omnivorous observation is 
everything*; but if we ask for something which we can 
know, it is like a vast mass. of indigestible material 
with every here and there a fibre or thread which we 
can assimilate. 

In this perplexity I was reduced to the last condition 
of mental despair ; and in default of finding anything 
which I could understand in nature, I was sufficiently 
humbled to learn anything which seemed to afibrd a 
capacity of being known. 

And the objects which came before me for this en- 
deavour were the simple ones which will be plentifully 
used in the practical part of this book. For I found 
tihat_the_o nly assert ion Lcould make about ^external 
objects, without ^bringing in unknown and unintel- 
ligible relations, was this : I could say how thnigs were 
arranged... If a stone lay between two others, that Vas 
a definite and intelligible fact, and seemed primary. As 
a stone itself, it was an unknown somewhat which one 
could get more and more information about the more 
one studied the various sciences. But granting that 
there were some things there which we call stones^ the 
way thqr were arranged was a siinple and obvious fact 
which could be easily expressed aiid easily remembered. 
And so in despair of being able to obtain any other 
kind of mental possession in the way of knowledge, I 
commenced to learn arrangements, and I took as Uie 
objects to be arranged certain artificial objects of a 
simple shape. I built up a block of cubes, and giving 
each a name I learnt a mass of them. 

Now I do not recommend this as a thing to be done. 
All I can say is that genuinely then and now it seemed 
mud seemstobetheonly kind of mental possession which 
can call kiiowledge. It is perfectly definite and 






1 



kome to thOT. A" I !~' " ^ ^^ i«ritably, by / 

Sl'^^'^'S^Siconc.P*- "^^ 
great minds have given us. oycction to an in- 

^I do not think it ought *« ^ J^^ J'^J^^on details. 

ambitious beginning ^^ „j^f 

And now I feel Uiat 1 n*^ «"J. ^ wUlhave the 
to the -tJ^^- °^J^;r^ «d buTldupoa our 
patience to go on, we wui ocgu. 

foundations. 



Apprehension of Nature. IiUMigence. 15 



CHAPTER II. 

Am«,EKSION or NATURE. INTELLIGENCE. STUDV 
OF ARRANGEMENT OR SHAPR 

Nature is that which is around us B«f .* • k 

Thus it is clear that it i« ««♦ T <>» naturt 

not a free gift- ''" ""^ e<»i ol an activity, 

TT.t'i^e"Urj:?^'. f. ^Pr'^nd'ng nature. 

•Set affin^S^'u^u *''* '"''**'^«* harmonies and the 
•ecret affinities which poetiy makes us feel • thJ/-^^ 

2^. theroi, the definite Wled^. oftaiu^Tfal ^ 
wludi the memory and reason are Siploy^. " 

i»to•:x*:Sh•:2t:;^^= "°''"° '^ «„, our mind. 

•notheiJS^riwJ^ tMy cannot undewtand one 
«d«S^l^*, ^^ «"« »V which he can 
^S^^t^^t^^ Aut out by. limitatiooof 



And thus our power of understanding nature depends 
on our own possession ; it is in virtue of some mental 
activity of our own that we can apprehend that outside 
activity which we call nature. And thus the training to 
enable us to approach nature with our minds will be 
some active process on our own part 

In the course of my experience as a teacher I have 
often been struck by the want of the power of reason 
displayed by pupils ; they are not able to put two and 
two together, as the saying goes, and I have been at 
some pains to investigate wherein this curious deficiency 
lies, and how it can be supplied. And I have found 
that there is in the curriculum no direct cure for it — ^the 
discipline which supplies it is not one which comes into 
school methods, it is a something which most children 
obtain in the natural and unsupervised education of their 
first contact with the world, and lies before any recog- 
nised mode of distinction. They can only understand 
in virtue of an activity of their own, and they have not 
had sufficient exercise in this activity. 

In the present state of education it is impossible to 
diverge from the ordinary routine. But it is always 
possible to experiment on children who are out of the 
common line of education. And I believe I am amply 
justified by the result of my experiments. 

I have seen that the same activity which I have 
found makes that habit of mind which we call intelli- 
gence in a child, is the source of our common and every- 
day rational intellectual work, and that just as the 
faculties of a child can be called forth by it, so also the 
powers of a man are best prepared by the same means^ 
but on an ampler scale. 

A more detailed . development of the practical work 
of Part II., would be the best training for the mind ot 
a child An extension of the work of that Part would 



i6 



A New Era of Thought. 



be the training which, hand in hand with observation and 
lecapitulation, would best develop a man's thought power. 

In order to tell what the activity is by the prosecution 
of which we can obtain mental contact with nature 
we should observe what it b which we say we ^ under- 
stand " in any phenomenon of nature which has become 
dear to us. 

When we look at a bright object it seems very dif« 
ferent from a dull one. A piece of bright steel hardly 
looks like the same substance as a piece of dull steel 
But the difference of appearance in the two is easily 
accounted for by the different nature of the surface In 
the two cases ; in the one all the irregularities are done 
away with, and the rays of light which fall on it are sent 
off again without being dispersed and broken up. In 
the case of the dull iron the rays of light are broken up 
and divided, so that they are not transmitted with 
intensity in any one direction, but flung off in all sorts 
of directions 

Here the difference between the bright object and the 
dull object lies in the arrangement of the particles on its 
surface and their influence on the rays of light 

Again, with light itself the differences of colour are 
explained as being the effect on us of rays of different 
rates of vibration. Now a vibration is essentially this, a 
scries of arrangements of matter which follow each 
other in a closed order, so that when the set has been 
run through, the first arrangement follows again. The 
whole theory of light is an account of arrangements 
of the particles In the transmitting medium, only the 
arrangements alter^are not permanent In any one 
characteristic, but go through a complete cycle' of 
varieties. 

Again, when the movements of the heavenly bodies 
are deduced from the theory of universal gravitation, 



Study of Arrangemmt or S/iape. 1 7 

what we primarily do is to take account of arrangement ; 
for the law of gravity connects the. movements which 
the attracted bodies tend to make with their disUnc^, 
that is, it shows how their movements depend on their 
arrangement And if gravity as a force is to be explained 
itself, the suppositions which have been put forward 
resolve it into the effect of the movements of small 
bodies ; that is to say, gravity, if explained at all, is 
explained as the result of the arrangement and altering 
arrangements of small particles. 

y^ain, to take the idea which proceeding from Goethe 
casts such an influence on botanical observation. 
Goethe (and also WolQ laid down that the parts of a 
flower were modified leaves— and traced the sUges and 
intermediate states between the ordinary green leaf and 
the most gorgeous petal or stamen or carpel, so un- 
like a leaf in form and function. 

Now the essential value in this conception is, that 
it enables us to look upon these different organs of a 
plant as modifications of one and the same organ— it 
enables us to think about the different varieties of the 
flower head as modifications of one single plant form. 
We can trace correspondences between them, and are 
led to possible explanations of their growth. And all 
this is done by getting rid of pistil and sUmen as separ- 
ate entities, and looking on them as leaves, and their 
parts due to different arrangement of the leaf structure. 
We have reduced these diverse objects to a common 
element, we have found the unit by whose arrangements 
the whole is produced. And in this department of 
- thought, as also to take another instance, in chemistry, 
to understand is practically this : we find units (leaves 
or atoms) combinations of which account for the results 
which we see. Thus we see that that which tlie mind 
essentially apprehends is arrangement 



I 



I thing «milar in nature to that which we ^hS^ 
««oge el«nent. into compounded ^1 ^ "^ "^ 
mi^. *• •*"*5: """S-n-'t in the active way we 



Study of ArrangcnuHt or oaape. ty 

If the elements are not alike then we are not study- 
ing pure arrangement; but our knowledge is affected by 
the compound nature of that with which we deal If 
the elements are all alike^ we have what we call units. 
Hence the discipline preparatory for the understanding 
of nature is the active arrangement pf like units. 

And this is very much the case with all educational 
processies; only the things chosen to arrange are in 
general words, which are so complicated and carry such 
a train of association that, unless the mind has already 
acquired a knowledge of arrangement, it is puzzled and 
hampered, and never gets a clear apprehension of what 
its work is. 

Now what shall we choose for our units ? Any unit 
would do ; but it ought to be a real thing — ^it ought to 
be something which can be touched and seen, not some- 
thing which no one has ever touched or seen, and which 
is even incapable of definition, like a ^ number.'' 

I would divide studies into two classes : those which 
create the faculty of arrangement, and those which use 
it and exercise it Mathematics exercises i^ but I do 
not think it creates it; and unfortunately, in mathe- 
matics as it is now often taught, the pupil is launched 
into a vast system of symbols— the whole use and 
meaning of symbols (namely, as means to acquire a 
clear grasp of facts) is lost to him. 

Of the possible units which will serve, I take the 
cube ; and I have found that whenever I took any other 
unit I got wrong, puzzled and lost my way. With the 
cube one does not get along very fast, but everything 
is perfectly obvious and simple, and builds up into a 
whole of which every part is evident 

And I must ask the reader to absolutely erase from 
his mind all desire or wish to be able to predict or 
assert anything about nature^ and he must please look 



AW^MV 



^^» OJ I/iOUg/tt. 



^^^r^"^ »-ag.natIo„-^, bring It into con" 



CHAPTER III. 

THE ELEMENTS OF KNOWLEDGE. 

There are two elements which enter into our know* 
ledge with respect to any phenomenon. 

If, for instance, we take the sun, and ask ourselves 
what we observe, we Qotice that it is a bright, moving 
body; and of these two qualities, the brightness and 
the movement, each seems equally predicable of the 
sun. It does move, and it is bright 
' Now further study discloses to us that there is a 
difference between these two affirmations. The motion 
of tlie sun in its diurnal course round the earth is only 
apparent ; but it is really a bright, hot body. 

Now of these two assertions which the mind naturally 
makes about the sun, one — ^that it is moving— depends 
on the relation of the beholder to the sun, the other is 
true about the sun itself. The observed motion depends 
on a fact affecting oneself and having nothing to do 
with the sun, while the brightness is really a quality of 
the sun itself. 

Now we will call those qualities or appearances which 
we notice in a body which are due to the particular 
conditions under which oneself is placed in observing 
it, the self elements ; those facts about it which are 
independent of the observer's particular relationship we 
will call the residual element Thus the sun's motion • 
is a self element in our thought of the sun, its brightness 
is a residual element ^ 



^y€w lira of Thouj^ht. 
For i«to,K^ SietS. K ?'«.'«»•<'«»' dement; 

they ^ ^srfhT^r ' Tk' '"^ •^'^^^^^ 

giving forth a e^tL! of ^^°? *^' **« «"> « 
•ndl^t ^^**'**'°^*"««y*n the form of heat 

«^.*^'U?n dt^J" PV-'"i the d«c„„io„ 
And it is J^r^. TT"^. " 'uffidently obvious. 

«3fd«aI'eIen.Sri^i;'J" for^'"^ '^ *^»* ««« 
By getting rid of the^S^eSe^T. *''*'*''■ ****"*''°"- 
• position in which we «n i ^7* P"* °'"^'''« ^n 
By getting rid of L lo'Kr"'' T'""" ^"««o««- 
tlie earth we .^1 °'^"' *="*"'*'' "ot'on round 

^ -id«z"«^ti:t^ri'^''i^'^<'-"-thiscon^ 

r^gen^TButrelrirSTt'd-'L?"'' **"''^ '^^ 
h« been dea«d of the sSrelLln J .>"^'^ * '"''j«^' 
to,have M them .ntAdu«^ aT JuhatT' "^ '^""' ' 
«Ity there was in eett.V«r^i\u .*,*''* er«'* *'«- 

WiU. regard toTSf^^^ ^^veXe "• 
. day about sdenfa-fic matte«^fhLr J^ *''* P'^nt 

wy«elf element, p^?^*?';? ^"^ "*>* »e«n to be 
e>«nent i^ that it, ^^.^ ^* '^"' *^"* * »«'f 

*waywithit AiuTTi, J" .**"•»*<> h*ve done 

• fl^d wScL'ke^c^' ::s ^u^i Jr '""^ ^ «•'« 

e^ent^faUd^^^^-^r^ ^^ '^"'' 
Now one of our serious piec^,f^^^,„ ^^^^^ 



7)i^ Elements of Knowledge. 



23 



rid of the self elements in the knowledge of arrange- 
ment. 

And the kind of knowledge which we shall try to 
obtain will be somewhat different from the kind of 
knowledge which we have about events or natural 
phenomena. In the large subjects which generally 
occupy the mind the things thought of are so compli- 
cated that every detail cannot possibly be considered. 
The principles of the whole are realized, and then at 
any required time the principles can be worked out 
Thus, with regard to a knowledge of the planetaiy 
system, it is said to be known if the law of movement 
of each of the planets is recognized, and their positions 
at any one time, committed to memory. It is not our 
habit to remember their relative positions with r^ard 
to one another at many intervals, so as to have an 
exhaustive catalogue of them in our minds. But with 
regard to the elements of knowledge with which we 
shall work, the subject is so simple that we may justly 
demand of ourselves that we will know every detaiL 

And the knowledge we shall acquire will be much 
more one of the sense and feeling than of the reason. 
We do not want to have a rule in our minds by which 
we can recall the positions of the different cubes, but 
we want to have an immediate apprehension of them. 
It was Kant who first pointed out how much of thought 
there was embodied in the sense impressions ; and it is 
this embodied thought which we wish to fomu 



CHAPTER IV. 



THEORY AND PRACTICE. 



Both in sdence and in morals there is an important 
distinction to be drawn between theory and practice. 
A knowledge of chemistry does not consist in the in- 
teUectdU appreciation of different theories and principles, 
but in being able to act in accordance with the facts 
of chemical combination, so that by means of the ap- 
pliances of chemistry practical results are produced. 
And so in morals— die theoretic acquaintance with the 
princtpfes of human action may consist with a marked 
6egrec of ignorance of how to act amongst other human 
beings. 

Now the use of the word ^ learn *' has been much 
restricted to merely theoretic studies. It requires to be 
enlarged to the scientific meaning. And to know, should 
include, practice in actual manipulation. 

Let us take an instance. We all know what justice 
is,<md any child can be taught to tell the difference 
between acting justly and acting unjustly. But it is a 
different thing to teach them to act with justice. Some« 
thing is done which affects them unpleasantly. They 
fed an impulse to retaliate. In order to see what justice 
demands they have to put their personal feeling on one 
side. They have to get rid of those conditions under 
which they apprehended the effects of the action at first, 
and they have to look on it from another point of view. 
Then they have to act in accordance with this view. 



T/Uary and Fractue. ^^ 

. ^ at-na— one an intellectual one of 

Now there are two ^^^PJr^^^ne of carrying out the 

undersUnding,one ^ P'^.^*^^^^^ demands intellU 

view. Neither is a f <>^*^ .^^JP' ^^^^^ „d this habit 

gence,theod.erJ.eform^^^^^^^ ^, ^^,.3,. 

can be inculcated ^y P^""^^^^ human nature is 
ment, by various ^^^f.!?^^ inculcated it touches 
constituted if the habit o^^^^^^^^ ^^^^^^ 

a part of the being. There is an 
W'e know but little of aJ^T-.^^J^^^^^^ ^^^ of 
«,y that there »«*;P*^J^^ ^^j^, ^f pleasurable 
momentary resentment a"° /" . . ^^.^^ 
or painful sensation, to which J"»J'J* " "J^^^iedgc of a 
S:>w little -^-^^^'^J,;Z^'rl^ZJi^^ of 
human being as a ^''y;'*"*'ne of these isolated 
human life. Nowandagam we see one of^^ ^ ^^^ 

beings bound up m ^^^^^^ And in all there 
covered Ph/^J^^^l^^k^S oHndwelling verdict of 
te this sense of «f fT^ ^Iv use such an expression. 
rvSJrofwSihVlrfrnot ^ . smgle individual 

^"^^th 'r:;:;t to i-^^-2r„e:.Tr^ 

take the -«* o^-^J^^j^^'SJe which » very good 
of many. There ^»y «3 J , ^^ but very bad 
justice from ^^^^'f^^^^l „S And if we 
justice from the po.n^o^^'.ewo^^^^ race, gifted with 

suppose an •B^'^J.^'r'^lerace.intheway for instance 
intelligence, and affecung the '^ ^ j„ o^er 

of causing »t«?J»? o'^»'; ^^Sd jXtake'^ stand- 
tojudgeitwithjusu^weshoua ^^^^^^ 

point outside the «ce of «en^>gem ^^^ ^^^ ^ 

,»y that this agency was ^ ^^ usa^M^. 

judge it with reference to it» effett on ou. 

"^^ ««ne word, which i«o(teoi«cd la contrat 



26 



New Era of Thought. 



with each other— ^oism and altruism ; and each seems 
to me unmeaning except as terms in a contrast. 

Let us Uke an insUnce. A boy has a bag of calces, 
and is going to enjoy them by himself. His parent 
stops him» and makes him set iip some stumps and 
b^n to learn to play cricket with another boy. The 
enjoyment rf the cakes is lost— he has given that up ; 
but aiker a little while he has a pleasure which is greater 
than that of cakes in solitude. He enters into the life 
of the game. He has given up, or been forced to give 
UR the pleasure he knew, and he has found a greater 
one. What he thought about himself before was that 
he liked cakes, now what he thinks about himself is 
that he likes cricket; Which of these is the true thought 
about himself? Neither, probably, but at any rate it 
is more near the truth to say that he likes the cricket 
If nowjwe use. the word self to mean that which a 
person knows of himself, and it is difficult to see what 
other meaning it can have, .his self as he knew It at first 
was thwarted, was given up, and through that he dis- 
covered his true self. And ^ain with the cricket ; he 
will make the sacrifice of giving that up, voluntarily or 
involuntarily, and will find a truer self stilL 

In general there is not much difficulty in making a 
bqy find out that he likes cricket ; and it is quite pos- 
sible for him to, eat his cakes first and learn to play 
cricket afterwards— the cricket will not come to him as 
a thwarting in any sense of what he likes better. But 
this ease in entering in to the pursuit only shows that . 
the boy's nature is already developed to the level of 
enjoying the game. The distinct moral advance would 
come in such a case when something which at first was 
hard to him to do was presented to him— and the hard- 
nesa, the unpleasantness is of a double kind, the giving 
up of a pursuit or indulgence to which he is accustomed* 



Theory and Practice. 



27 



and the exertion of forming the habits demanded by 
the new pursuit . . 

Now it is unimportant whether the renunciation is 
forced or willingly Uken. But as a general rule it may 
be laid down, that by giving up his own desires as he 
feek them at the moment, to the needs and advantage 
of those around him, or to the objects which he finds 
before him demanding accomplishment, a human being 
passes to the discovery of his true self on and on. The 
process is limited by the responsibilities which a man 
finds come upon him. . ,, 

The method of moral advance is to acquire a practical 
knowledge; he must first see what the advantage of 
some one other than himself would be, and then hc^ ^^ 
must act in accordance with that view of things. Then 
having acted and formed a habit, he discovers a response^ 
in himself. He finds that he really cares, and that his 
former limited life was not really himself. His body and 
the needs of his body, so far as he can observe them, 
externally are the same as before ; but he has obtained 
an inner and unintellcctual, but none the less real, 
apprehension of what he is. , 

Thus altruism, or the sacrifice of egoism to others, is 
followed by a truer egoism, or assertion of self, and 
this process flashed across by the transcendent lights 
of religion, wherein, as in the sense of justice and duty, 
untold depths in the nature of man are revealed entirely 
unexpressed by the intellectual apprehension which we 
have of him as an animal frame of a very high d^ee 
of development, is the normal one by which from child-; 
hood a human being develops into the full response 
bilitiesofaman. 

Now both in science and in conduct there are self, 
dements. In science, getting rid of the self elements j 
means a truer apprehension of the facts about one ; in ; 



38 



New Era of Thought. 



' .^tl^?^ Ji"^ *• "^^ element, mean, obtain- 
"2 a truer knowledge of what we are-in the wav of 

^i-TuTSir '^^•^ '^« ^■^-^ 

In aT^;"^' Pitoidtoe to My .cioi(i8c «„™™ 

pot tlvone, into practice can be Mfcly sati,fied-!«w 

l-.'i^?*^*?.*' "**"■>"'•*''»" "eveiything; we must 
.^^«d h!"' "^ ~"**'''«"'^ oth^^weTaaT 

Su,^ *rf»« «• due to the wrongncM of our con! 



CHAPTER V. 

knowledge: self-elements. 

I MUST now go with somewhat of detail into the special 
subject in which these general truths will be exhibited. 
Everything I have to say would be conceived much 
more clearly by a very little practical manipulation. 

But here I want to put the subject in as general a 
light as possible, so that there may be no hindrance to 
the judgment of the reader. 

' And when I use the word " know," I assume some- 
thing else than the possession of a rule, by which it can 
be said how facts are. By knowing I mean that the 
facts of a subject all lie in the min^ ready to come out 
vividly into consciousness when the attention is directed 
on them. Michael Angelo knew the human frame, he 
could tell every little fact about it ; if he chose to call 
up the image, he would see mentally how each muscle 
and fold of the skin lay with regard to the surhninding 
parts. We want to obtain a knowledge as good as 
Michael Angelo's. There is a great difference between 
Michael Angelo and us ; but let that difference be ex- 
pressed, not in our way of knowing, but in the difference 
between the things he knew and the things we know. 
We take a very simple structure and know it as abso> 
lutely as he knew the complicated structure of the 
human body. 

And let us take a block of cubes ; any number will do, 
but for convenience sake let us take a set of twenty-seven • 



30 



New Era of Thought, 



cubes put together so as to form a lai^e cube of twenty- 
•even parts. And let each of these cubes be marked 
•o as to be fecognired,and let each have a name so that 

ILT* ♦^•'trr* ^ ^^ '•* "* »"PP°»« «»»* *« have 
I«mit this block of cubes sp that each one Is known- 

ttat IS to say, Its position in the block is known and its 
rdahon to the other blocks. 

Now having obtained this knowledge of the block as 
It stands Ml front of us, let us ask ourselves if there is 
any self element present in our knowledge of it 

And there is obviously this self element present We 
have Icamt the cubes as they sUnd in accordance with 
our own convenience in putting them up. We put the 

and we distincdy conceive the lower ones as supportine 

S^l^'^TK^^TM^r,*^" ^ "^ ~PPort has\!othi„| 
todo with the^bkHdc of cubes itself, it depends on thf 
condibons under which we come to apprehend the block 
of cubes. It depends on our position on the surface of 
the earth, whereby gravity is an all important (actor in 
our experience. In fact our sight has got so accustomed 
to take gravity into consideration in its view of things 
thatwhen we lookat a landscape or object with ou^ 
head upside down we do not see it inverted, but we 
SHPennduce on the direct sense impressions our know- 
ledge^the action of gravity,and obtainaview differing 
joy Irttle from what we see when in an upright posi- 

It will be found that every fact about the cubes has 
involved w it a reference to up and dowa It is by 
being above or below that we chiefly remember where 
Jecubes^e. But above «kJ betow fa . rS^ ZuS 
depadM su^y on gravity. If it were not for gravity 

^^.,^Zr^.J^ interehangeable tc^Z 
ttad of opreiring a difference of marked importance 



KnowUdf^ : Seif-eUments. 



3' 



to us under our conditions of existence. Now we put 
. "being above" or "being below" into the cubes them- 
selves and feel it a quality in them — it defines their 
position. But this above or below really comes from 
the conditions in which we are. It is a self element, and 
as such, to obtain a true knowledge of the cubes we 
must get rid of it 

And now, for the sake of a process which will be ex- 
plained afterwards, let us suppose that we cannot move 
the block of cubes which we have put up. Let us keep 
it fixed. 

In order to learn how it is independent of gravity the 
best way would be to go to a place where gravity has 
virtually ceased to act ; at the centre of the earth, for 
instance, or in a freely falling shell 

But this is impossible, so we must choose another way. 
Let us, then, since we cannot get rid of gravity, see 
what we have done already. We have learnt the cubes, 
and however they are learnt, it wHl be found that there 
is a certain set of them round which the others are 
mentally grouped, as being on the right or left, above 
or below. Now to get our knowledge as perfect as we 
can before getting rid of the self element up and down, 
we have to take as central cubes in our mind different 
sets again and again, until there are none which are 
primary to us. 

Then there remains only the distinction of some being 
above others. Now this can only be made to sink out 
of the primary place in our thoughts by reversing the 
relatioa If we turned the block upside down, and 
learnt it in this new position, then we should learn the 
position of the cubes with regard to each other with 
that element in thqn, which comes from the action of 
gravity, reversed. And the true nature of the arrange- 
ment to which we added something in virtue of our 



32 



New Era of Thought. 



sensation of up and down, would become purer and more 
isolated in our minds. 

We have, however, supposed that the cubes are fixed. 
Then, in onler to learn them, we must put up another 
block showing what thqr would be like in the supposed 
new position. We then take a set of cubes, models of 
the original cubes, and by consideration we can put 
them in such positions as to be an exact model of what 
the bhxrk of cubes would be if turned upside down. 

And here is the whole point on which the process 
dependsL We can tell where each cube would come, 
but we do not hum the block in this new position. I 
draw a distinction between the two acts, ^ to tell where 
it would be," and to "know.'' Telling where it would 
be is the preparation for knowing. The power of as* 
signing the positions may be called the theory of the 
block? The actual knowledge is got by carrying out 
tiie theoiy practically, by putting up the blocks and 
becoming able to realize - without effort where each 
one is. 

It is not enough to put up the model blocks in the 
reverse position. It is found that this up arid down 
is a very obstinate element indeed, and a good deal 
of work is requisite to get rid of it completely. But 
when it is got rid of in one set of cubes, the faculty 
is formed of .appreciating shape independently of the 
paitkrular parts which are above or below on first ex- 
amination. We discover in our own minds the faculty 
of appreciating tiie facts of position independent of 
gravity and itsinfiuence on us. I have found a very 
great difference in different minds in this respect To 
some it is easy, to some it is hard. 

And to use our old instance^ the discovery of this 
capacity is Uke the discovery of a love of justice in tiie 
beiog who has forced himself to act justiy. It is a 



Knowledge: Self-elemefUs. 



33 






capacity for being able to take a view independent of 
the conditions under which he is placed, and to feel in 
accordance with that view. There is, so far as I know, 
no means of arriving immediately at this impartial ap- 
preciation of shape. It can only be done by, as it were, 
extending our own body so as to include certain cubes, 
and appreciating then the relation of the other cubes to 
those. And after this, by identifying ourselves with 
other cubes, and in turn appreciating the relation of the 
other cubes to these. And the practical putting up of 
tiie cubes is the way in which this power is gained. It 
springs up with a repetition of the mechanical acts. Thus 
there are tiiree processes, ist, An apprehension of what 
the position of the cubes would be. 2nd, An actual put- 
ting of them up in accordance with that apprehension. 
3rd, The springing up in the mind of a direct feeling of 
what the block is, independent of any particular pre- 
sentation. 

Thus the self element of up and down can be got rid 
of out of a block of cubes. 

And when even a little block is known like this, the 
mind has gained a great deal. 

Yet in the apprehension and knowledge of the block 
of cubes with the up and down relation in them, there 
is more than in the absolute apprehension of them. For 
there is the apprehension of their position and abo of 
the effect of gravity on them in their position. 

Imagine ourselves to be translated sgddenly to 
another part of the universe, and to find there intelli- ' 
gent beings, and to hold conversation with them. If 
we told them that we came from a world, and were to 
describe the sun to them, saying that it was a bright, 
hot body which moved round us, they would reply: 
You have told us something about the sun, but you have 
also told us something about yourselves. 



34 



A New Era of ThonghL 



Thus in the apprehension of the sun as a body moving 
round us there is more than in the apprehension of it as 
not moving round, for we really in this case apprehend 
two things — ^the sun and our own conditions. But for 
the purpose of further knowledge it is most important 
tl)at the more abstract knowledge should be acquired. 
The self element introduced by the motion of the earth 
must be got rid of before the true relations of the solar 
system can be made out 

And in our block of cubes, it will be found that feel- 
ings about arrangement, and knowledge of space, which 
ar^ quite unattainable with our ordinary view of posi- 
tion, become simple and clear when this discipline has 
been gone through. 

And there can be no possible mental harm in going 
through this bit of training, for all that it comes to is 
kxiking at' a real thing as it actually is— turning it 
round and over and learning it from every point of 



■^ 



: \ 



\.-' 



CHAPTER VI. 

FUNCTION OF MIND. SPACE AGAINST METAPHYSICS, 
SELF-UMITATION AND ITS TEST. A PLANE WORLD. 

We now pass on to the question : Are there any other 
self elements present in our knowledge of the block of 

cubes.? 

When we have learnt to free it from up and down, is 
there anything else to be got rid of? 

It seems as if, when the cubes were thus learnt, we had 
got as abstract and impersonal a bit of knowledge as 
possible. 

But, in reality, in the relations of the cubes as we thus 
apprehend them there is present a self element to which 
the up and down is a mere trifle. If we think we have 
got absolute knowledge we are indeed walking on a 
thin crust in unconsciousness of the depths below. 

We are so certain of that which we are habituated to, 
we are so sure that the world is made up of the me- 
chanical forces and principles which we familiarly deal 
with, that it is more of a shock than a welcome surprise 
to us to find how mistaken we were. 

And after all, do we suppose that the facts of distance 
and size and shape are the ultimate facts of the world- 
is it in truth made up like a machine out of mechanical 
parts ? If so, where is there room for that other which 
we know— more certainly, because inwardly— that reve- 
rence and love iR4iich make life worth having? No; 
these mechanical relations are our means of knowin 



36 



A Niw Era of Thought. 



about the worid ; they are not reality itself, and their 
primaiy place in our imaginations is due to the famili- 
ari^ which we have with them»and to the peculiar limi- 
tations under which we are. 

But I do not for a moment wish to go in thought be- 
yond physical nature-^I do not suppose that in thought 
we can. To the mind it. is only the body that appears, 
and all that I hope to do is to show material relations, 
mechanism, arrangements. 

But much depends on what kind of material relations 
we perceive outside us. A human being, an animal and 
a machihe are to the mind all merely portions of matter 
arranged in certain ways. But the mind can give an 
exhaustive account of the machine, account fairly well 
for jthe animal, while the human being it only defines 
externally, leaving the real knowledge to be supplied by 
cyther faculties. ' 

. But we must not under-estimate the work of the mind, 
for it is only by the observation of and thought about the 
bodies with which we come into contact that we know 
human beings. It is the faculty of thought that puts us 
in a position to recognize a souL 

And. so^ toOp about the universe^it is only by correct 
thought about it that we can perceive its true moral 
nature 

And- it will 'be found that the deadness which we 
^ ascribe to the external world is not really there, but is 
I pat in by us because of our own limitations. It is really 
I the self elements in our knowledge which make us talk 
I of medianical necessity, dead matter. When our limt- 
! tations fall, we behold the spirit of the world like we be- 
; hold the spirit erf* a friend— something which is discerned 
! ift and through the material presentation of a body to 

Our thought means are sufBdent at present to show 



Space against Metaphysics. 



37 



us human souls ; but all except human beings is, as far 
as science is concerned, inanimate. One self element 
must be got rid of from our perception, and this will be 
changed. 

The one thing necessary is, that in matters of thinking 
we will not admit anything that is not perfectly clear, 
palpable and evident. On the mind the only conceiv- 
able demand is to seek for facts. The rock on which so 
many systems of philosophy have come to grief is the 
attempt to put moral principles into nature. Our only 
duty is to accept what we find. Man is no more the 
centre of the moral world than he is of the physical 
world. Then relegate the intellect to its right position 
of dealing with facts of arrangement — it can appreciate 
structure — and let it simply look on the world and report 
on it. We have to choose between metaphysics and* 
space thought In metaphysics we find lofty ideals-— 
principles enthroned high in our souls, but which reduce 
the world to a phantom, and ourselves to the lofty spec* 
tators of an arid solitude On the other hand, if we 
follow Kant's advice, we use our means and find realities 
linked together, and in the physical interplay of forces 
and connexion of structure we behold the relations i> 
between spirits — those dwelling in man and those above J, 
him. 

It is difficult to explain this next self element that has 
to be removed from the block of cubes ; it requires a 
little careful preparation, in fact our language hardly 
affords us the means. But it is possible to approach in* 
directly, and to detect the self-eleqaent by means of an 
analogy. 

If we suspect there be some condition affecting our- 
selves which make us perceive things not as theyare, 
but falsely, then it is possible to test the matter by mak- 
ing the supposition of other beings subject to certain 



38 



A New Era of Thought. 



I 



cooditioDSy and then examining what the effect on their 
experience would be of these conditions. 

Thus if we make up the appearances which would 
present themselves to a being subject to a limitation or 
condition, we shall find that this limitation or condition, 
when unrecognized by him, presents itself as a general 
law of his outward world, or as properties and qualities 
of the objects external to him. He will, moreover, find 
certain operations possible, others impossible, and the 
boundaiy line between the possible and impossible will 
depend quite as much on the conditions under which he 
is as on the nature of the operations. 

And if we find that in our experience of the outward 
world there are analogous properties and qualities of 
matter, analogous possibilities and impossibilities, then 
it will show to us that we in our turn are under analo- 
gous limitaticos^ and that what we perceive as the ex- 
ternal world is both the external world and our own 
conditions. And the task before us will be to separate 
the twa Now the problem we take up here is this — ^to 
separate the self elements from the true fact To separ- 
ate them not merely as an outward theory and intelligent 
s|>prdiension, but to separate them in the consciousness 
itself! so that our power of perception is raised to a 
higher level We find out that we are under limitations. 
Our next step is to so familiarize ourselves with the real 
aspect of things^ that we perceive like beings not under 
ourlimitatk>ns. Or more truly, we find that inward 
soul which itself not subject to these limitations, is 
awakened to its own natural action, when the verdicts 
conveyed to it through the senses are purged of the self 
eiemeots introduced by the senses. 

Eveiytfaing depends on this — Is there a native and 
^MMitaneons power of apprehension, which springs into 
activityiHien we take the trouble to present to it a view 



Self'limUatiofi attd its Test. 



I 






39 ' 

from which the self elements are eliminated? About f 
this every 'one must judge for himself. But the pro- | 
cess whereby this inner vision is called on is a de- ^ 
finite one. 

And just as a human being placed in natural human 
relationships finds in himself a spontaneous motive 
towards the fulfilment of them, discovers in himself a 
being whose motives transcend the limits of bodily self- 
regard, so we should expect to find in our minds a power 
which is ready to apprehend a more absolute order of 
fact than that which comes through the senses. 

I do not mean a theoretical power. A theory is al- 
ways about it, and about it only. I mean an inner view, 
a vision whereby the seeing mind as it were identifies 
iuclf with the thing seen. Not the tree of knowledge, 
but of the inner and vital sap which builds up the tree 
df knowledge. 

And if this point is settled, it will be of some use in 
answering the question : What are we ? Are we then 
bodies only ? This question has been answered in the 
negative by our instincts. Why should we despair of a 
rational answer? Let us adopt our space thought and 
develop it 

The supposition which we must make is the followmg. 
Let us imagine a smooth surface— like the surface of a 
table ; but let the solid body at which wc arc looking be 
very thin, so that our surface is more like the surface of 
a thin sheet of metal than the top of a table. 

And let us imagine small particles, like particles of 
dust, to lie on this surface, and to be attracted down- 
wards so that they keep on the surface. But let us sup- 
pose them to move freely over tjie surface. Let them 
never in their movements rise one over the other ; let 
them all singly and collectively be close to the surface. 
And let us suppose all sorts of attractions and repulsions • 



40 



A New Era of Thought. 



between these atoms, and let them have all kincls of 
movements like the atoms of our matter have. 

Then there may be conceived a whole world, and 
various kinds of beings as formed out of this matter. 
The peculiarity about this world and these beings would 
be, that neither the inanimate nor the animate members 
of it would move away from the surface. Their move- 
ments would all lie in one plane, a plane parallel to and 
vciy near the surface on which they are. 

And if we suppose a vast mass to be formed out of 
these atoms, and to lie like a great round disk on the 
surface; compact and cohering closely together, then this 
great disk would afford a support for the smaller shapes, 
which we may suppose to be animate beings! The 
smaller shapes would be attracted to the great disk, but 
would be arrested at its rim. They would tend to the 
centre of the disk, but be unable to get nearer to the 
centre than its rim. 

Thus, as we are attracts to the centre of the earth, 
but walk on its surface, the beings on this disk would be 
attracted to its centre, but walk on its rim« The force 
of attraction which they would feel would be tho attrac- 
tion of the disk. The other force of attraction, acting 
perpendicularly to the plane which keeps them and all the 
matter of their worid to the surface, they would know 
nothing about* For they cannot move either towards this 
force or away from it ; and the surface is quite smooth, 
so that they feel no friction in their movement over it 

Now let us realize clearly one of these beings as he 
proceeds along the rim of his worid. Let us imagine 
him in the form of an outline of a human being, with no 
thickness except that of the atoms of his world. As to 
the mode in which he walks, we must imagine that he 
proceeds by springs or hops, because there woulcj be no 
room for his limbs to pass each other. 



Self-limitation and its Test. 



41 



Imagine a lai^e disk on the table before you, and a 
being, such as the one described, proceeding round it 
Let there be small movable particles surrounding him, 
which move out of his way as he goes along, and let 
these serve him for respiration ; let them constitute an 
atmosphere. 

Forwards and backwards would be to such a being 
direction along the rim — the direction in which he was 
proceeding and its reverse. 

Then up and down would evidently be the direction 
away from the disk's centre and towards it Thus back- 
wards and forwards, up and down, would both lie in the 
plane in which he was. 

And he would have no other liberty of .movement 
except these. Thus the words right and left would have 
no meaning to him. All the directions in which he 
could move, or could conceive movement possible, would 
be exhausted when he had thought of the directions 
along the rim and at right angles to it, both in the plane. 

What he would call solid bodies, would be groups of 
the atoms of his world cohering together. Such a mass 
of atoms would, wc knpw, have a slight thickness ; 
namely, the thickness of a single atom. But of this ho 
would know nothing. He would say, ^ A solid body 
has two dimehsions-^height (by how much it goes away 
from the rim) and thickness (by how much it lies along 
the rim).'' Thus a solid would be a two-dimensional 
body, and a solid would be bounded by lines. Lines 
would be all that he could see of a solid body. 

Thus one of the results of the limitations under which 
he exists would be, that he would say, ** There are only 
two dimensions in real things." 

In order for his world to be permanent; we must 
suppose the surface on which he is to be very compact, . 
compared to the particles of his matter ; to be very 



4^ 



-r/ New Era of TtioughL 



rigid ; and, if he \% not to observe it by the friction of 
matter moving on it, to be very smooth. And if it is 
wy compact with regard to his matter, the vibrations of 
the surface must have the effect of disturbing the portions 
of his matter, and of separating compound bodies up 
into simpler ones. 

Another consequence of the Umitadon under which 
this being Iies» would be the foUowing :— If we cut out 




from the comers of a piece of paper two triangles, ABC 
and A'FC, and suppose them to be reduced to such 
^ thinness that they are capable of being put on to the 
imaginary suriace, and of being observed by the flat, 
being like other bodies known to him ; he will, after 
studying the bounding lines, which are all that he can see 
w touch, come to the conclusion that they are equal and 
similar in evciy respect ; and he can conceive the one 
occupying the same space as the other occupies, without 
its being altered in any way; 

IC however, instead of putting down these triangles 
wtotiic surface on which the supposed being lives, as 
••wwn in Fig. i, we first of all tuni one of them over, 



A Plane World. 



43 



and then put them down, then the plane-being has pre- 
sented to him two triangles, as shown in Fig. 2. 

And if he studies these, he finds that they are equal 
in size and similar in e^txy respect But he cannot 
make the one occupy the same space as the other one ; 
this will become evident if the triangles be moved about 
on the surface of a table. One will not lie on the same 
portion of the table that the other has marked out by 
lying on it 

Hence the plane-being by no means could make the 
one triangle in this case coincide with the space occupied 
by the other, npr would he be able to conceive the one 
as coincident with the other. 

The reason of this impossibility is, not tliat the one 
cannot be made to coincide, but that before having been 
put down on his plane it has been turned round. It 
has been turned, using a direction of motion which the 
plane-being has never had any experience of, and which 
therefore he cannot use in his mental work any more 
than in his practical endeavoura 

Thus, owing to his limitations, there is a certain line 
of possibility which he cannot overstep. But this line 
does not correspond to what is actually possible and 
impossible. It corresponds to a certain condition affect* 
ing him, not affecting the triangle. His saying that it 
is impossible to make the two triangles coincide, is an 
assertion, not about the triangles, but about himsel£ 

Now, to return to our own world, no doubt there are 
many assertions which we make about the external 
world which are really assertions about ourselves. And 
we have a set of statements which are precisely similar 
to those which the plane-being would make about his 
surroundings. 

Thus, he would say, there are only two independent 
directions ; we say there are only three. 



44 



A New Era of Thought. 



He would say that solids are bounded by lines ; we 
say that solids are bounded by planes. 

Moreover, there are figures about which we assert 
exactly the same kind of impossibility as his plane-being 
did about the triangles in Fig, 2. 

We know certain shapes which are equal the one to 
the other, which are exactly similar, and yet which we 
cannot make fit into the same portion of space, either 
practically or by imagination. 

If we look at our .two hands we see this clearly, 
though the two hands are a complicated case of a very 
commoA fact of shape. Now, there id. one way in 
which the right hand and the left hand may practically 
be brought into likeness. If we take the r^ht-hand 
glove and the left-hand glove, they will not fit any more 
than the right hand will coincide with the left hand. 
But if we turn one glove inside out, then it will fit Nowf 
to suppose the same thing done with the solid hand as 
is done with the glove whai it is turned inside out, we 
must suppose it, so to speak, pulled through itsel£ If 
the hand were inside the glove all the time the glove 
was being turned inside out, then, if such an operation 
were possible, the right hand would be turned into an 
exact model of the left hand Such an operation is 
ioipossible. But curiously enough there is a precisely 
similar operation which, if it were possible; would, in a 
plane, turn the one triangle in Fig. 2 into the exact 
copy of the other. 

Look at the triangle in Fig. 2, A B C, and imagine 
the point A to move into the interior of the triangle and 
to pass through it, carrying after it the parts of the lines 
AB and AC to which it is attached, we should have 
filially a triangle ABC, which was quite like the other 
of the two triangles A' B" C in Fig. 2. 
Thus we know the operation which produces the 



A Plane World. 



45 



result of the " pulling through " is not an impossible one 
when the plane-being is concerned. Then may it not be 
that there is a way in which the results of the impossible 
operation of pulling a hand through could be performed i 
The question is an open one. Our feeling of it being 
impossible to produce this result in any way, may be 
because it really is impossible, or it may be a useful bit 
of information about ourselves. 

Now at this point my special work comes in. If there 
be really a four-dimensional world, and we are limited 



A, A A 






to a space or three-dimensional view, then tither we are 
absolutely three-dimensional with no experience at all 
or capacity of apprehending four-dimensional facts, or 
we may be, as far as our outward experience goes, so 
limited ; but we may really be four-dimensional beings 
whose consciousness is by certain undetermined con« 
ditions limited to a section of the real space. 

Thus we may really be like the plane-beings mentioned 
above, or we may be in such a condition that our percep* 
tions, not ourselves, are so limited The question is one 
which calls for experiment 

We know that if we Uke an animal, such as a dog 



46 



A New Era of TltouglU. 



or cat, we can by careful training, and by using rewards 
and punishment, make them act in a certain way, in 
certain defined cases, in accordance with justice; we 
can produce the mechanical action. But the feeling 
of justice will not be aroused ; it will be but a mere 
outward conformity. But a human being, if so trained, 
and seeing others so actmg, gets a feeling of justice. 

Now, if we are really four-dimensional, by going 
through those acts which correspond to a four-dimen- 
'sional experience (so far as we can), we shall obtain an 
apprehension of four-dimensional existence — not with 
the outward eye^ but essentially with the mind. 
^ And after a number of years of experiment which were 
entirely nugatory, I can now lay it down as a verifiable 
I fact, that by taking the proper steps we can feel four- 
dimensional existence^ that the human being somehow, 
and in some way, is not simply a three-dimensional 
being — in what way it is the province of science to 
discover* All that I shall do here is, to put forward 
certain suppositions which, in an arbitrary and forced 
manner, give an outline of the relation of our body to 
four-dimensional existence^ and show how in our minds 
we have fiiculties by which we recognise it 



CHAPTER VII. 

SELF ELEMENTS IN OUR CONSCIOUSNESS. 

It is often taken for granted that our consciousness of 
ourselves and of our own feelings has a sort of direct 
and absolute value. 

It is supposed to afford a testimony which does not 
require to be sifted like our consciousness of external 
events. But in reality it needs far more criticism to be 
applied to it than any other mode of apprehension. 

To a certain degree we can sift our experience of 
the external world, and divide it into two portions. 
We can determine the self elements and the realitiea 
But with r^[ard to our own nature and emotions, the 
discovery which makes a science possible has yet to be 
made. 

There are certain indications, however, springing from 
our observation of our own bodies, which have a certain 
dq^ree of interest 

It is found that the processes of thought and feeling 
are connected with the brain. If the brain is disturbed, 
thoughts, sights, and sounds come into the conscious- 
ness which have no objective cause in the external 
world. Hence we may conclusively say that the human 
being, whatever he is, is in contact with the brain, and 
through the brain with the body, and through the body 
with the external world. 

It is the structures and movements in the brain which ' 

47 - 



48 



A New Era of Thought. 



the httman being perceives. It is by a structure in the 
biain that he apprehends nature, not immediately. 
The most beautiful sights and sounds have no effect 
on a human being unless there is the faculty in the 
brain of taking them in and handing them on to the 
consciousness. 

Hence, clearly, it is the movements and structure of 
the minute portions of matter forming the brain which 
the consciousness perceives. And it is only by models 
and representations made in the stuff of the brain that 
the mind knows external changes. 

Now^ottr brains are well furnished with models and 
representations of the facts and events of the external 
world* 

fiut a most important fact still requires its due weight 
to be laid upon it 

These models and representations are made on a very 
minute scale — ^the particles ot brain matter which form 
images and representationa are beyond the power of the 
microsoope in their minuteness. Hence the conscious- 
ness primarily apprehends the movements of matter of 
a degree of smallness which is beyond the power of 
observation in any other way. 

Heniw we have a means of observing the movements 
oC the minute portions of matter. Let us call those 
portions of the brain matter which are directly instru- 
mental in making representations of the external world 
— let us call them brain molecules. 

Now, these brain molecules are very minute portions 
of matter indeed ; generally they are made to go 
throi^h movements and form structures in such a way 
as to represent the movements and structures of the 
external world of masses around us. 

But it does not follow that the structures and move- 
irfiich they perform of their own nature are 



Self Elements in our Consciousness. 49 

identical with the movements of the portions of matter 
which we see around us in the world of matter. 

It may be that these brain molecules have the power 
of four-dimensional movement, and that they can go 
through four-dimensional movements and form four- 
dimensional structures. 

If so, there is a practical way of learning the move- 
ments of the very small particles of matter— by observ- 
ing, not what we can see, but what we can think. 

Fori suppose these small molecules of the brain were 
to build up structures and go through movements not 
in accordance with the rule of representing what goes 
on in the external world, but in accordance with their 
own activity, then they might go through four-dimen- 
sional movements and form four-dimensional structures. 

And these movements and structures would be ap- 
prehended by the consciousness along with the other 
movements and structures, and would seem as real as 
the others— but would have no correspondence in the 
external world. 

They would be thoughts and imaginationSi not ob- 
servations of external facts. 

Now, this field of investigation is one which requires 
to be worked at 

At present it is only those structures and movements 
of the brain molecules which correspond to the realities 
of our three-dimensional space which are in general 
worked at consistently. But in the practical part of 
this book it will be found that by proper stimulus the 
brain molecules will arrange themselves in structures 
representing a four-dimensional existence. It only 
requires a certain amount of care to build up mental 
models of higher space existences. In fact, it is pro- 
bably part of the difficulty of forming three-dimensional 
brain models, that the brain molecules have to be Umited 



so 



A Nov Era of Thought. 



m their own freedom of motion to the requirements of 
the liouted space in which our practical daily life is 
carried 00. 

N0U.^Yv my own pan I should say that all those confusions in 
itmembering which come from an Imago taking the place of the 
orighul mental model-as, for instance, the difficulty in lemember- 
lag which way to turn a screw, and the numerous cases of inures 
in dMN^ht tiansference— may be due to a toppling over in the 
hmm, looTHiimensionalwise, of the structures formed— which 
stmctnies would be absohitely safe from being turned into Image 
siractnres II the brain molecules moved only thiee^iimensional- 
wise 

It is' remaricable how in science "explaining - means 
the reference of the movements and tendencies to 
movement of the masses about us to the movements 
and tendencies to movement of tiie minute portions of 
matter. 

Thui the behaviour of gaseous bodies— die pressure 
which tfaqr exert, the laws of tiieir cooling and inter- 
mixture are exphdned by tracing the movements of the 
trcrjr mimite particles of which tiiey are composed 






CHAPTER VIII. 

RELATION OF LOWER TO HIGHER SPACE. 
OF THE iETHER. 



THEORY 



At this point of our inquiries the best plan is to turn 
to the practical work, and try if the faculty of thinking 
in higher space can be awakened in the mind. 

The general outline of the method is the same as that 
which has been described for getting rid of the limita- 
tion of up and down from a block of cubes. We sup- 
posed that the block was fixed ; and to get the sense of 
what it would be when gravity acted in a different way 
with regard to it, we made a model of it as it would be 
under the new circumstances. We thought out the 
relations which would exist ; and by practising this new 
arrangement we gradually formed the direct appre- 
' hension. 

And so witii higher-space arrangements. We cannot' 
put them up actually, but we can say how they would 
look and be to the touch from various sides. And we 
can put up the actual appearances of them, not alto- 
gether, but as models succeeding one another ; and by 
contemplation and active arrangement of these different 
views we call upon our inward power to manifest itself. 

In preparing our general plan of work, it is necessary 
to make definite assumptions with regard to our world, 
our universe, or we may call it our space, in relation to 
the wider universe of four-dimensional space. 

What our relation to it may be, is altogether un* . 
determined The real relationship will require a great 



k 



5« 



A New Era of Thought. 



deal or study to apprehend, and when apprehended will 
seem as natural to us as the position of the earth among 
the other planets does to us now. 

But we have not got to wait for this exploration in 
order to commence our work of higher-space thought, 
for we know definitely that whatever our real physical 
relationship to this wider universe may be, we are practi- 
cally in exactly the same relationship to it as the 
creature we have supposed living on the surface of a 
smooth sheet is to the world of threefold space. 

And this relationship of a surface to a solid or of a 
solid, as we conjecture, to a higher solid, is one which 
we often find in nature. A surface is nothing more nor 
less than the relation between two things. Two bodies 
touch each other. The surface is the relationship of one 
to the other. 

Again, we see the surface of water. 

Thus our solid existence may be the contact of two 
four-dimensional existences with each other; and just as 
sensation of touch is limited to the surface' of the body, 
so sensation on a larger scale may be limited to this 
solid surface. 

And it is a fact worthy of notice, that in the surface 
of a fluid different laws obtain from those which hold 
throughout the mass. There are a whole series of facts 
which sure grouped together under the name of surface 
tensions, which are of great importance in physics, and 
by whidi the behaviour of the surfaces of liquids is 
governed. 

And it may well be that the laws of our universe are 
the surface tensions of a higher universe 

But these expressions, it is evident, afford us no practi- 
cal basis for investigation. We must' assume something 
more definite, and because more definite (in the absence 
of details drawn from experience)^ more arUtrary. 



Relation of Lower to Higher Space. 53 

And we will assume that the conditions under which 
we human beings are, exactly resemble those under 
which the plane-beings are placed, which have been 
described. 

This forms the basis of our work ; and the practical 
part of it consists in doing, with regard to higher 
space, that which a plane-being would do with regard 
to our space in oider to enable himself to realize what 
it was. 

If we imagine one of these limited creatures whose 
life is cramped and confined studying the facts of space 
existence, we find that he can do it in two ways. He 
can assume another direction in addition to those which 
he knows ; and he can, by means of abstract reasoning, 
say what would Uke place in an ampler kind of space 
than his own. All this would be formal work. The 
' conclusions would be abstract possibilities. 
. The other mode of study is this. He can take some 
of these facts of his higher space and he can ponder 
over them in his mind, and can make up in his plane 
world those different appearances which one and the 
same solid body would present to him, and then he may 
try to realize inwardly what his higher existence is. 

Now, it is evident that if the creature is absolutely 
confined to a two-dimensional existence, then anything 
more than such existence will always be a mere abstract 
and formal consideration to him. 

But if this higher-space thought becomes real to him, 
if he finds in his mind a possibility of rising to it, then 
indeed he knows that somehow he is not limited to his 
apparent world. Everything he sees and comes into 
contact with may be two-dimensional ; but essentially, 
somehow, himself he is not two-dimensional merely. 

And a precisely similar piece of work Is before us. 
Assuming as we must that our outer experience is 



54 



A Niw Era of Thwgla. 



limited to thfee-dimensional spacer we shall make up 
the appearances which the very simplest higher bodies 
would preseot to US| and we shall gnulually arrive at a 
more than merely formal and abstract appreciation of 
them. We shall discover in ourselves a faculty of ap- 
pirhension of higher space similar to that which we have 
of space And thus we shall discover, each for himself, 
that, limited as his senses are, he essentially somehow 
is not limited. 
^ The mode and method in which this consciousness 
will be made general, Is the same In which the spirit of 
an armx is formed. 

The individuals enter into the service from various 

motives, but each ai^d all have to go through those 

movements and actioiis which correspond to the unity 

; of a whole formed out of different naembers. The inner 

; apprehension which lies in each man of a participation 

[in a life wider than that of his individual body, is 

^ awakened and responds ; and the active spirit of the 

' army is formed. So with regard to higher space, this 

i faculty of apprehending intuitively four-dimensional 

> relationships will be taken up because of its practical 

j use. Individuals will be practically employed to do it 

{ by society because of the larger faculty of thought 

! which it gives. In fact, this higher-space thought means 

: as an affair of ipental training simply the power of ap- 

I prehending the results arising from four independent 

I causes. It means the power of dealing with a greater 

' number of detaiU 

And when this Caculty of higher-space thought has 
'been formed, then the faculty of apprehending that 
higher existence in which men have part,, will come 
into being. 

It is necessary to guard here against there being 
ascribed to this higher«space thought any other than 



Relation of Lower to Higlur Space. 55 

an intellectual value. It has no moral value whatever. 

lU only connexion with moral or ethical considerations 

is the possibility it will afford of recognizing more of 

the facts of the universe than we do now. There is a 

gradual process going on which may be described as 

Ae getting rid of self elements. This process is one of 

knowledge and feeling, and either may be independent 

of the other. At present, in respect of feeling, we arc 

much further on than in respect to understanding, and 

the reason is very much this : When a self element has 

been got rid of in respect of feeling, the new appre- 

hension Is put into practice, and wc live it «nto our 

organization. But when a self element has been got rid 

of intellectually, it is' allowed to remain a matter ot 

theory, not vitelly entering into the mental structure of 

individuals. ir^i^«.^«#. 

• Thus up and down was discovered to be a self element 
more than athousand years ago ; but, except as a matter 
of theory, we are perfect barbarians in this respect up to 
the present day. , , 

We have supposed a being living in a plane world, 
that is, a being of a very small thickness in a direction 
perpendicular to the surface on which he is. ^ 

Now, if we are situated analogously with regard to 
an ampler space, there must be some element in our 
experience corresponding to each element m the plane- 
being's experience. ^ . , 

And it is interesting to ask, in the case of the plane- 
being, what his opinion would be with respect to the 
surface on which he was. 

He would not recognize it as a surface with which 
he was in contact ; he would have no idea of a motion 
away from it or towards it . 

But he would discover its existence by the fact ftat 
movements were transmitted along it By its vibrating 



56 



A New Era of Tiumgkt. 



and quivering, it would impart movement to the par- 
ticles of nmtter lying on it 

Hence^ he would consider this surface to be a medium 
lying between bodies, and penetrating them. It would 
appear to him to have no weight but to be a powerful 
means of transmitting vibrations. Moreover, it would 
be unlike any other substance with which he was 
acquainted, inasmuch as he could never get rid of 
it However perfect a vacuum be made, there would 
be in this vacuum just as much of this unknown me- 
dium as there was before. 

Moreover, this surface would not hinder the move* 
mcnt of the particles of matter over it Being smooth, 
matter wouM slide freely over it And this would seem 
to him as if matter went freely through the medium* 
* Then he would also notice the fact that vibrations 
of thislroediuni would tear asunder portions of matter. 
The plane surface, being very compact, compared to 
the masses of matter on 'it, would, by its vibiations, 
shake them into their component parts. 

Hence he would have a series of observations which 
tended to show that this medium was unlike any or. 
dinary matter with which he was acquainted Although 
matter' passed freely through it, still by its shaking it 
could tear matter in pieces. These would be very 
difficult properties to reconcile in one and the same 
substance. Then it is weightless, and it is everywhere. 

It might well be that he would regard the suppo- 
sition of there being a plane surface^ on which he was^ 
as a preferable one to the hypothesis of this curious 
medium ; and thus he might obtain a proof of his limi- 
tations from his observations. 

Now, is there anything in our experience which 
corresponds to'this medium which the plane-being geU 
toobMTve? 



Theory of the ^Iher. 



57 



Do we suppose the existence of any medium through caK 
which matter freely moves, which yet by its vibrations 
destroys the combinations of matter— some medium 
which is present in every vacuum, however perfect 
which penetrates all bodies, and yet can never belaid 
hold of? 

These are precisely observations which have been 
made. 

The substance which possesses all these qualities is 
called the aether. And the properties of the aether are 
a perpetual object of investigation in science. 

Now, it is not the place here to go into detoils, as 
all wc want is a basis for work ; and however arbitrary 
it may be, it will serve if it enables us to investigate 
the properties of higher space. 

We will suppose, then, that we are not in, but on the 
ether, only not on it in any known direction, but that 
the new direction is that which comes in. The aether 
is a smooth body, along which we slide, being distant 
from it at every point about the thickness of an atom ; 
or, if we take our mean distance, being distant from 
it by half the thickness of an atom measured in this 
new directioit 

Then, just as in space objects, a cube^ for instance, 
can stend on the surface of a table, or on the surface 
over which the plane-being moves, so on the aether can 
stand a higher solid. 

All that the plane-being 9Ct^ or touches of a cube, 
is the square on which it rests. 

So all that we could see or touch of a higher solid 
would be that part by which it stood on the aether ; 
and this part would be to us exactly like any ordinary 
solid body. The base of a cube would be to the 
plane-being like a square which is to him an ordinary 
solid. 



58 



A New Era of Thought. 



Now, the two ways, in which a plane-being would 
apprehend a solid body, would be by the successive 
appearances to him of it as it passed through his plane ; 
and also by the different views of one and the same solid 
body which he got by turning the body over, so that 
different parts of its surface come into contact with his 
plane* 

And the practical work of learning to think in four- 
dimensional spacer is to go through the appearances 
which one and the same higher solid has. 

Often, in the course of investigation in nature, we 
come across objects which have a certain similarity, and 
yet which are in parts entirely different The work of the 
mind consists in forming an idea of that whole in which 
they cohere^ and of which they are simple presentations. 
The work of forming an idea of a higher solid is the 
most Simple And most definite of all such mental 
operations. 

If we imagine a plane world in which there are 
objects which correspond to our sun, to the planets, and, 
in fact, to all our visible universe, we must suppose a 
surface of enormous extent on which great disks slide, 
these disks ,being worlds of various orders of magni- 
tude. • 

•These disks would some of them be ceiitral, and hot, 
like our sun; lound them would circulate other dbks^ 
like our planets. 

And the systems of sun and planets must be con- 
ceived as moving with great velocity over the surface 
which bears them alL 

And the movements of the atoms of these worlds 
will be the course of events in such worlds. As the 
atoms weave together, and form bodies altering, be- 
coming, and ceasing, so will bodies be formed and 
disappear. 



Theory of the j£t/ter. 



59 



And the plane which bears them all on its smooth 
surface will simply be a support to all these movements^ 
and influence them in no way. 

Is to be conscious of being conscious of being hot, 
the same thing as to be conscious of being hot ? It is 
not the same. There is a standing outside, and objecti- 
vation of a state of mind which every one would say in 
the first state was very different from the simple con- 
sciousness. But the consciousness must do as much in 
the first case as in the second. Hence the feeling hot 
is very different from the consciousness of feeling hot 

A feeling which we always have, we should not be 
conscious of— a sound always present ceases to be heard. 
Hence consciousness is a concomitant of change, that 
is, of the contact between one state and another. 

If a being living on such a plane were to investigate 
the properties, he would have to suppose the solid to 
pass through his plane in order to see the whole of 
its surface. Thus we may imagine a cube resting on 
a table to begin to penetrate through the table. If the 
cube passes through the surface, making a dean cut all 
round it, so that the plane-being can come up to it and 
investigate it, then the different parts of the cube as it 
passes through the plane will b^ to him squares, which 
he apprehends by the boundary lines. The cut which 
there is in his plane must be supposed not to be noticed, 
he must be able to go right up to the cube without hin- 
drance, and to touch and see that thin slice of it which 
is just above the plane. 

And so, when we study a higher solid, we must sup- 
pose that it passes through the aether, and that we only 
see that thin three-dimensional section of it which is 
just about to pass from one side to the other of the 
ether. 

When we look on a solid as a section of a higher' 



6o 



A Neuf Era of Thought. 



solid, we have to suppose the aether broken through, 
only we must suppose that it runs up to the edge of the 
body which is penetrating it, so that we are aware of 
no breach of continuity. 

The surface of the xther must then be supposed to 
have the properties of the surface of a fluid; only, of 
courscb it is a solid three-dimensional surface^ not a two- 
dimensional surface. 



CHAPTER IX. 

ANOTHER VIEW OF THE iGTHER. MATERIAL AND 
iCTHERIAL BODIES. 

We have supposed in the case of a plane world that the 
surface on which the movements take place is inactive, 
except by its vibrations. It is simply a smooth support. 

For the sake of simplicity let us call this smooth 
surface * the aether ^ in the case of a plane world. 

The aether then we have imagined to be simply a 
smooth, thin sheet, not possessed of any definite struc- 
ture, but excited by real disturbances of the matter on 
It into vibrations, which carry the effect of these dis- 
turbances as light and heat to other portions of matter. 
Now, it is possible to take an entirely different view of 
the aether in the case of a plane world. 

Let us imagine that, instead of the aether being a 
smooth sheet serving simply as a support, it is de- 
finitely marked and grooved. 

Let us imagine these grooves and channels to be very 
minute, but to be definite and permanent 

Then, let us suppose that, instead of the matter which 
slides in the aether having attractions and repulsions of 
its own, that it is quite inert^ and has only the properties 
of inertia. 

That is to say, taking a disk or a plane world as a 
specimen, the whole disk is sliding on the aether in 
virtue of a certain momentum which it has, and certain 
portions of its matter fit into the grooves in the aether, 
and move along those grooves. 

The size of the portions is determined by the sixe of 



62 



A New Era of Thought. 



the groovesL And let us call those portions of matter 
which occupy the breadth of a groove, atoms. Then it 
is evident that the disk sliding along over the ether, its 
atoms will move according to the arrangement of the 
grooves over which the disk slides. If the grooves at 
any one particular place come close together, there will . 
be a condensation of matter at that place when the 
disk passes over it ; and if the grooves separate^ there 
will be a rarefaction of matter. 

If we imagine five particles, each slipping along in its 
own groove, if the particles are arranged in the form of 
a regular pentagon, and the grooves are parallel, then 
these five particles^ moving evenly on, will maintain 
their positions with regard to one another, and a body 
would exist like a pentagon, lasting as long as .the 
groves /emaincd parallel 

But if, after some distance had been traversed by the 
disk, and these five particles were brought into a region 
where one of the grooves tended away from the others, 
the shape of the pentagon would be destroyed, it would 
become some irrq^ar figure. And it is easy to see 
that if the grooves separated, and other grooves came 
in amongst them, along which other portions of matter 
were sliding, that the pentagon would disappear as an 
isblated body, that its constituent matter would be 
separated, and that its particles would enter into other 
shapes as constituents of them, and not of the original 
pentagon. 

Thus, in cases of greater complication, an elaborate 
structure may be supposed to be formed, to alter, and to 
pass away ; its origin, growth, and decay being due^ not 
to aoy independent motion of the particles constituting 
It^ but to the movement of the disk whereby its portions 
of matter were brought to rq[k>ns where there was a 
particiilar disposition of tlie grooves. 



Another View of the /Ether. 



63 



Then the nature of the shape would really be deter- 
mined by the grooves, not by the portions of matter 
which passed over them — they would become manifest 
as giving rise to a material form when a disk passed 
over them, but they would subsist independently of the 
disk ; and if another disk were to pass over the same 
grooves, exactly the same material structures would 
spring up as came into being before. 

If we make a similar supposition about our aether 
along which our earth slides, we may conceive the 
movements of the particles of matter to be determined, 
not by attractions or repulsions exerted on one another, 
but to be set in existence by the alterations in the 
directions of the grooves of the sether along which 
they are proceeding. 

If the grooves were all parallel, the earth would pro- 
ceed without any other motion than that of its path in 
the heavens. 

But with an alteration in the direction of the grooves, 
the particles, instead of proceeding uniformly with the 
mass of the earth, would begin to move amongst each 
other. And by a sufficiently complicated arrangement 
of grooves it may be supposed that all the movements 
of the forms we see around us are due to interweaving 
. and variously disposed grooves. 

Thus the movements, which any body goes through, 
would depend on the arrangement of the aethereal 
grooves along which it was passing. As long as the 
grooves remain grouped together in approximately the 
same way, it would maintain its existence as the same 
body ; but when the grooves separated, and became in-* 
volved with the grooves of other objects^ this body 
would cease to exist separately. 

Thus the separate existences of the earth might con- 
ceivably be due to the disposition of those parts of the 



04 



A New Era of Thought. 



setfacr over which the earth passed. And thus any 
object would have to be separated into two parts, one 
the aethereal form, or modification which lasted, the 
other the material particles which, coming on with 
blind momentum, were directed into such movements as 
to produce the actual objects around us. 

In this way there would be two parts in any organism, 
the material part and the aethereal part There would 
be the material body, which soon passes and becomes 
indistinguishable froni any other material body, and the 
aethereal body which remains. 

Now, if we direct our attention to the material body, 
we see the phenomena of growth, decay, and death, the 
coming and the passing away of a living being, isolated 
during his existence, absolutely merged at his death into 
the common storehouse of matter. 

Buf if we regard the aethereal body, we find something 
different We find an organism which is not so abso- 
lutely separated from the surrounding organisms — an 
organism which is part of the aether, and which is linked 
to other aethereal organisms by its \txy substance — an 
organism between which and others there exists a unity 
incapable of being broken, and a common life which is 
rather marked than revealed by the matter which passes 
cnrer it The aethereal body moreover' remains per- 
manently when the nuterial body has passed away. 

The correspondences between the aethereal body and 
the life of an organism such as we know, is rather to be 
found in the emotional r^on than in the one of out- 
ward observation. To the aethereal form, all parts of it 
are equally one ; but part of this form corresponds to 
the future of the nuterial being, part of it to his past 
Thiia» care for the future and regard for the past would 
be the way In which the material being would exhibit 
tiie iini^ of the aethereal body^ which is both his past, 



Material and /Ethereal Bodies. 



his present, and his future. That is to say, suppose the / 
aethereal body capable of receiving an injury, an injury I 
in one part of it would correspond to an injury in a l 
man's past ; an injury in another part, — that which the 
material body was traversing, — would correspond to an | 
injury to the man at the present moment ; injury to the 
ethereal body at another part, would correspond to 
injury coming to the man at some future time.. And 
the self-preservation of the aethereal body, supposing it 
to have such a motive, would in the last case be thei 
motive of regarding his own future to the man. And| 
inasmuch as the man felt the real unity of his aetherealj 
body, and did not confine his attention to his material, 
body, which is absolutely disunited at every moment 
from its future and its past — ^inasmuch as he apprehended 
his aethereal unity^ insomuch would he care for his future 
'welfare, and consider it as equal in importance to his 
present comfort The correspondence between emotion 
and physical fact would be, that the emotion of regai^ 
corresponded to an undiscerned aethereal unity. And 
then also, just as the two tips of two fingers put down 
on a plane, would seem to a plane-being to be two com- 
pletely different bodies, not connected tc^ether, so one 
and the same aethereal body might appear as two 
distinct material bodies, and any regard between the 
two would correspond to an apprehension of their 
aethereal unity. In the supposition of an aethereal body, 
it is not necessary to keep to the idea of the rigidity and 
permanence of the grooves defining the motion of the 
matter which, passing along, exhibits the material body. 
The aethereal body may have a life of its own, relations 
with other aethereal bodies, and a life as full of vidssi* 
tudes as that of the material body, which in its total 
orbit expresses in the movements of matter one phase . 
in the life of the aethereal body. 



66 



A New Era of Thought 



But there are certain obvious considerations which 
prevent any serious dwelling on these speculations— they 
are only introduced here in order to show how the con- 
ception of higher space lends itself to the representation 
of certain indefinite apprehensionsi— such as that of the 
essential unity of the race,— and aifords a possible clue 
to correspondences between the emotional and the 
physical life* 

The whole question of our relation to the aether has 
to be settled. That which we call the aether is far more 
probably the surface of a liquid, and the phenomena we 
observe due to surface tensions. Indeed, the physical 
questions concern us here nothing at all It is easy 
enough to make some supposition which gives us a 
standing ground to discipline our higher-space percep- 
tion ; and when that is trained^ we shall turn round and 
look at the facts. 

The conception which we shall form of the universe 
will undoubtedly be as different from our present one, 
9A the Copemican view diifers from the more pleasant 
view of a wide immovable earth beneath a vast vault 
Indeed, any conception of our place in the universe will 
be UMMre agreeable than the thought of being on a 
qMnning ball, kicked into space without any means of 
communication with any other inhabitants of the 
universe 



CHAPTER X. 



HIGHER SPACE AND HIGHER BEING. PERCEPTION AND 
INSPIRATION. 

In the instinctive and sense perception of man and 
nature there is all hidden, which reflection afterwards 
brings into consciousness. 

We are conscious of somewhat higher than each 
individual man when we look at men. In some, this 
consciousness reaches an extreme pitch, and becomes 
a religious apprehension. But in none is it otherwise 
than instinctive. The apprehension is sufficiently defi- 
• nite to be certain. But it is not expressible to us in 
terms of the reason. 

Now, I have shown that by using the conception of 
higher space it is easy enough to make a supposition 
which shall show all mankind as physical parts of one 
whole. Our apparent isolation as bodies from each 
other is by no means so necessary to assume as it 
would appear. But, of course, a supposition of that 
kind is of no value, except as showing a possibility. 
If we came to examine into the matter closely, we 
should find a natural relationship which accounted for 
our consciousness being limited as at present it is. 

The first thing to be done, is to organize our higher- 
space pereeption, and then look. We cannot tell what 
external objects will blend together into the unl^ of a 
higher being. But Just as the riddle of the two hands 
becomes clear to us from our first Inspection of higher 
space, so will there grow before our eyef greater unities, 
and greater surprises. 

«7 



68 



A Ntw Era of Thought, 



I 



We have been subject to a limitation of the most 
absunl character. Let us open our eyts and see the 
factSL 

Now, it requires some training to open the eyes. 
For many years I worked at the subject without the 
sl^test success. All was mere formalism. But by 
adopting the simplest means, and by a more thorough 
knowledge of spacer the whole flashed dear. 

Space shapes can only be symbolical of four-dimen« 
sional shapes ; and if we do not deal with space shapes 
directly, but only treat them by ajrmbols on the plane — 
as in analytical geometry— we are trying to get a per- 
ception of higher space through symbols of symbols, 
and the task is hopeless. But a direct study of space 
leads us to the knowledge of higher space. And with 
the knowledge of higher space there come into our ken 
boundfess possibilities. All those things may be real, 
whereof saints and philosophers have dreamed. 

Lodcing on the fact of life^ it has become clear to 
the human mind, that justice, truth, purity, are to be 
sought — that they are principles which it is well to 
serve. And men have invented an abstract devotion 
to tb^e, and all comes together in the grand but vague 
conceptkm of Duty. 

• But all these thoughts are to those which spring up 
before us as the shadow on a bank of clouds of a great 
mountain is to the mountain itselil On the piled-up 
clouds falls the shadow— vast, imposing, but dark, colour- 
lessi If the beholder but turns, he beholds the mountain 
itseli^ towering grandly with verdant pines, the snowline, 
and the awful peaks. 

So all these conceptions are the way in which now, 
with vision confined, we apprehend the great existences 
of the universe Instead of an abstraction, what we 
have to serve is a reality, to which even our real things 



Higher Spaa and Higher Being. 69 

are but shadows. We are parts of a great being, in 
whose service, and with whose love, the utmost demands 
of duty are satisfied. 

How can it not be a struggle, when the claims of 
righteousness mean diminished life,— even death,— to 
the individual who strives ? And yet to a clear and 
more rational view it will be seen that in his extinction 
and loss, that which he loves,— that real being which 
is to him shadowed forth in the present existence of 
wife and child,— that being lives more truly, and in iu 
life those he loves are his for ever. 

But, of course, there are mistakes in what we con- 
sider to be our duty, as in everything else ; and this is an 
additional reason for pursuing the quest of this reality. 
For by the rational observance of other material bodies 
than our own, we come to the conclusion that there 
are other beings around us like ourselves, whom we 
apprehend in virtue of two processes— the one simply 
a sense one of observation and reflection— the other a 
process of direct apprehension, 

Now, if we did not go through the sense process of 
observation, we might, it is true, know that there were 
other human beings around us in some subtle way— in 
some mesmeric feeling ; but we should not have that 
organized human life which, dealing with the things of 
the world, grows into such complicated forms. We . 
should for ever be good-humoured babies— a sensuous, 
aflectionate kind of jelly-fish. 

And just so now with reference to the high intelli- 
gences by whom we are surrounded. We feel them, 
but we do not realize them. 

To realize them, it will be necessaiy to develop our 
power of perception. 

The power of seeing with our bodily eye is limited to. \ 
the three-dimensional section. 



TO 



A New Era of Tfumgla. 



But I have shown that the inner eye is not thus 
limited ; that we can organize our power of seeing in 
higher spacer and that we can form conceptions of 
realities in this higher space, just as vre can in our ordi- 
nary space. 

And this aflbrds the groundwork for the perception 

and study of these other beings than man. Just as some 

mechanical means are necessary for the apprehension 

, of our fellows in space, so a certain amount of me- 

/ chanical education is. necessary for the perception of 

I h^her beings in higher space. 

I Let ifi turn the current of our thought right round ; 
instead of seeking after abstractions, and connecting our 
observations by ideas, let us train our sense of higher 
space and build up conceptions of greater realities, more 
absolute existences. 

It is* really a waste of time to write or read more 
generalit]e&' Here is the grammar of the knowledge of 
higher being— let us leamr it, not spend time in specu- 
lating as to whither it will lead us. 

Yet one thing more. We are, with reference to the 
higher things of life, like blind and puzzled children. 
We know that we are members of one body, limbs of 
one vine ; but we cannot discern, except by instinct and 
feeling* what that body is, what the vine is. If to know 
it would take Away our feeling, then it were well never 
to know it But fuller knowledge of other human beings 
does not take away our love for them ; what reason is 
there then to lupposc that a knowledge of the higher 
existences would deaden our feelings \ 

And then, again, we each of us have a feeling that we 
ourselves have a right to exist We demand our own 
perpetuation. No man, I believe, is capable of sacri- 
ficing his life to any abstract idea ; in all cases it is the 
consciousness of conUct with some being that enables 



Higlur Space and Higher Being. 71 

him to make the last human sacrifice. And what we 
can do by this study of higher space, is to make this 
consciousness, which has been reserved for a few, the 
property of all Do we not all feel that there is a limit 
to our devotion to abstractions, none to beings whom 
we love. And to love them, we must know them. 

Then, just as our own individual life is empty and 
meaningless without those we love, so the life of the 
human race is emp^ and meaningless without a know- 
ledge of those thi^t surround it And although to some 
an inner knowledge of the oneness of all men is vouch- 
safed, it remain^ to be demonstrated to tlie many. 

The perpetual struggle between individual interests 
and the common good can only be solved by merging 
both impulses in a love towards one being whose- life 
lies in the fulfilnient of each. . 

And this search, it seems to me, affords the needful 
supplement to the inquiries of one with whose thought 
I have been very familiar, and to which I retuni again, 
after having abandoned it for the purely materialistic 
views which seem forced upon us by the facts of science. 

All that he said seemed to me unsupported by fact, 
unrelated to what we know. 

But when I found that my knowledge was merely an 
empty pretence, that it was the vani^ of being able to 
predict and foretell that stood to me in the place of an 
absolute apprehension of fact — when all my intellectual 
possessions turned to nothingness, then I was forced 
into that simple quest for fact, which, when persisted in • 
and lived in^ opcni out to the thoughts like a flower to 
the life-giving sun. 

It is indeed a far safer course, to believe that which 
appeals to us as noble, than simply to ask what is true ; 
to take that which great minds have given, tlian to de- 
mand that our puny ones should be satisfied * But I 



^2 



A New Era of T/iougkt. 



suppose there is some good to soqne one in the seep- 
^f\ ticism and struggle ojf those who cannot follow in the 
■^ -^ safer courM. 
>^ The thoughts of the inquirer to whom I allude may 



y> . \ roughly be stated thus : — 



He saw in human life the working out of a great pro- 
^ cess^ in the toil and strain of our human history he saw 
"" the becoming of man. There is a defect whereby we fall 
^ i short of the true measure of our being, and that defect 
«^ is made good in the course of history, 
r It is owing to that i^efect that we perceive evil ; and 

^ in the 4)erception of evil and suffering lies our healing, 
I for we shall be forced into that path at last, after trying 
a every other, which is the true one 

And this^ the history of the redemption of man, b 
what he saw in all the scenes of life ; each most trivial 
occunwice was great and significant in relation to this. 

And, further, he put forward a definite sUtement with 
r^ard to this defect, this .lack of true being, for it lay, 
he said, in the self-centredness of our emotions, in the 
limitatk>n of them to our bodily selves. He looked for 
a time when, driven from all thoughts of our own pain 
or pleasure, good or evil, we should say, in view of the 
. miseries of pur fellow-creatures, Let me be anyhow, use 
my body and my mind in any way, so that I serve 

And this, it jeems to me^ is the true aq>iration ; for, 
just as a note of music flings itself into the march of the 
mefedy, and, losing itself in it, is used for it and lost as 
a separate being, so we should throw these lives of ours 
as freely into the service of~whom ? 

Here comes the dtfliculty. Let it be granted that we 
should have no self-rights, limit our service in no way. 
Still the question comes. What shall we serve ?. 

It is far happier to have some co n crete object to 
mkAdtk we are devoted, or to be bound up in the cease- 



Higlur Space and Htg/ier Being. 73 

less round of active life, wherein each day presents so 
many necessities that we have no room for choice 

But besides and apart from all these there comes to 
some the question, "What does it all mean?'' To others, 
an unlovable and gloomy aspect is presented, wherein 
their life seems to be but used as a material worthless in 
itself and ungifted with 'any dignity or honour; while 
to others again, with the love of those they love comes 
a cessation of all personal interest in life^ and a dis- 
appointment and feeling of valuelessness. 

And in all these cases some answer is needed. And 
here human duty ceases. We cannot make objects to 
love We can make machines and worics of art, but 
nothing which directly excites our love To give us 
that which rouses our love, is the duty of one higher 
than ourselves. 

And yet in one respect we have a du^-^we must 
look. 

What good would it be, to surround us with objects 
of loving interest, if we. bury our rq^ards in ourselves 
and will not see ? 

And does it not seem as if with lowered eyelids, till 
only the thinnest slit was open, we gazed persistently, 
not on what is, but on the thinnest conceivable section 
of it? 

Let it be granted that our right attitude is, so to 
devote ourselves that there is no question as to what we 
will do or what we will not do, but we aro perfectly 
obedient servants. The question is. Whom aro we to 
serve ? 

It cannot be each individual, for their claims are 
conflicting, and as often as not there is more need of 
a master than of a servant Moreover, the aspect of our 
fellows does not always excite love, which is the only . 
possible inducer of the right attitude of service If we 



74 



A New Era of Thought. 



do not lov€^ we can only serve for a self motive, because 
it is in some way good for ourselves. 

J ' Thus it seems to me that we are reduced to this : our 

I only doty is to look for that which it is given us to love. 

'*^ But this looking is not mere gazing. To know, we 
must act 

> Let any one try it. He will find that unless he 
! goes through a series of actions correspo/iding to his 
! knowledge, he gets merely a theoretic and outside view 
I of any factSL The w;ay to know is this : Get somehow 

r* a means erf telling what your perceptions would be if 
you knew, and act in accordance with those perceptions. 

La Thus, with regard to a fellow-creature, if we knew him 
we should feel what his feelings are. Let us then learn 
his feelings, and act as if we had them. It is by the 
practical work of satisfying his needs that we get to 
knowliim. 

Then, may-be, we love him ; or perchance it is said 
we may find that through him we have been brought 
into contact with one greater than him. 

This is our duty- to know-r-to know, not merely 
theoretically, but practically ; and then, when we know, 
we have done our part ; if there is nothing, we cannot 
supply it All we have to do is to look for realities. 
. We must not take this view of education— that we are 
horribly pressed for time, and must learn, somehow, a 
knack of saying how things must be, without looking at 
theoL 

But rather, we must say that we have a long time— all 
our lives, in which we will press facts closer and closer 
to our minds ; and we begin by learning the simplest 
There is an kiea in that home of our inspiration— the 
(act that there are certain mechanical processes by 
which men ean acquire merit This is perfectly true. 
It is by mechanical processes that we become different ; 



Perctption and Ifispiratton. 



75 



and the science of education consists largely in sys- 
tematizing these processes. 

Then, just as space perceptions are necessary for the 
knowledge of our fellow-men, and enable us to enter 
into human relationships with them in all the orgs^nized 
variety of civilized life, so it is necessary to develop t 
our perceptions of higher space, so that we can appre- [ 
hend with our minds the relationship which we have to | 
beings higher than ourselves, and bring our instinctive ' 
knowledge into clearer consciousness. ^ 

It appears to me self-evident, that in the particular 
disposition of any portion of matter, that is, in any 
. physical action, there can be neither right nor wrong ; 
the thing done is perfectly indifferent 

At the same time, it is only in things done that we 
come into relationship with the beings about us and 
higher than us. Consequently, in tlie things we do lies 
the whole importance of our lives. 

Now, many of our impulses are directly signs of a 
relationship in us to a being of which we are not imme- 
diately conscious. The feeling of love, for instance, is 
always directed towards a particular individual ; but by 
love man tends towards the preservation and improve- 
ment of his race ; thus in the commonest and most 
universal impulses lie his relations to higher beings than 
the individuals by whom he is surrounded. Now, along 
with these impulses are many instincts of a modifying 
tendency; and, being altogether in the dark as to the 
nature of the higher beings to whom we are related, it is 
difficult to say in what the service of the higher beings 
consists, in what it does not The only way is, as in 
every other pre-rational department of life, to take the 
verdict of those with the most insight and inspiration. 

And any striving against such verdicts^ and discontent * 
with them, should be turned into energy towards finding 



76 



A New Era of Thought. 



out exactly what relation we have towards these higher 

beings by the study of Space. 
^ Human life at present is an art constructed in its 
^ regulations and rules on the inspirations of those who 
/ love the undiscemed higher beings, of which we are a 

part. They love these higher beings, and know their 



But our perceptions are coarser; and it is only by 
labour and toil that we shall be brought also to see, and 
then lose the restraints that now are necessary to us in 
the fulness of love* 

£?tactly what relationship there is towards us on the 
part of these higher beings we cannot say in the least 
We cannot even say whether there is more than humanity 
before the highest ; and any conception which we form 
now must use the human drama as its only possible 
mcxle of presentation. 

But that there is such a relation seems clear ; and the 
ludicrous manner, in which our perceptions have been 
limited, is a sufficient explanation of why they have not 
been scientifically apprehended. 

The modc» in which an apprehension of these higher 
beings or being is at present secured, is as follows ; and 
it bears a striking analogy to the mode by which the 
.self is cut out of a block of cubes. 
*^ When we «tudy a block of cubes, we first of all learn 
it, by starting from a particular cube, and learning how 
all the others come with regard to that All the others 
are right or left, up or down, near or far, with regard to 
that particular cube. And the line of cubes starting 
from this first one^ which we take as the direction in 
which we look^ is, as it were, an axis about which the 
rest of the cub^ are grouped We learn the block with 
ngud to this, axis, so that we can mentally conceive 
the dispositioii of every cube as it comes regarded from 



Perception and Inspiration. 



n 



one point of view. Next we suppose ourselves to be in 
another cube at the extremity of another axis ; and, 
looking from this axis, we learn the aspects of all the 
cubes, and so on. 

Thus we impress on the feeling what the block of 
cubes is like from every axis. In this way we get a 
knowledge of the block of cubes. 

Now, to get a knowledge of humanity, we must feel 
with many individuals. Each individual is an axis as 
it were, and we must regard human beings from many 
different axes. And as, in learning the block of cubes, 
muscular action, as used in putting up the block of 
cubes, is the means by which we impress on the feeling 
the different views of the block; so^ with r^ard to 
humanity, it is by acting with regard to the view of each 
individual that a knowledge is obtained. That is to say, 
' that, besides sympathizing with each individual, we must 
act with regard to his view ; and acting so, we shall feel 
his view, and thus get to know humanity from more than 
one axis. Thus there springs up a feeling of humanity, 
and of more. 

Those who feel superficially with a great many people, 
are like those learners who have a slight acquaintance 
with a block of cubes from many points of view. Those fk 
who have some deep attachments, are like those who '' 
know them well from one or two points of view. 

Thus there are two definite paths— one by which the 
instinctive feeling is called out and developed, the other 
by whidh we gain the faculty of rationally apprehending 
and learning the higher beings. 

In the one way it is by the exercise of a sympathetic* 
and active life ; in the other, by the study of higher 
space. J 

Both should be followed ; but the latter way is more . 
accessible to those who are not good. For we at any 



78 



A New Era of Thought. 



rate have the industry to go through mechanical opera- 
tionsy and know that we need something. 

And after all, periiaps, the diflerence between the good 
and the rest of us, lies rather in the former being aware. 
There is something outside them which draws them to 
it» which they see while we do not 

There is no reason, however, why this knowledge 
shouM not become demonstrable fact Surely, it is only 
by becoming demonstrable fact that the errors which 
have been necessarily introduced into it by human 
weakness will fall away from it 

T^e rational knowledge will not replace feeling, but 
will form the vehicle by which the facts will be presented 
to our consciousness. Just as we learn to know our 
fellows by watching their deeds»— but it is something 
beyond the mere power of observing them that makes 
us regard them, — ^so the higher existences need to be 
known ; and, when known, then there is a chance that 
in the depths of our nature they will awaken feelings 
towards them like the natural response of one human 
being to another. 

And when we reflect on what surrounds us, when we 
think that the beauty of fruit and flower, the blue depths 
of the sky, the majesty of rock and ocean,— all these are 
- but the chance and arbitrary view which we have of true 
'bdng,— then we can imagine somewhat of the glories that 
await our coming. How set out in exquisite loveliness 
are all the budding trees and hedgerows on a spring day 
—from here^ where they almost sing to us in their near- 
ness^ to .whcare^ in the distance^ they stand up delicately 
distant and distinct in the amethyst ocean of the air I 
And there^ quiet and stately, revolve the slow moving 
sun and the stars of the night All these are the frag- 
mentary views which we have of great beings to whom 
we are related, to whom we are linked, did we but realize 



Perception and Inspiration. 



79 



it, by a bond of love and service in close connexions of 
mutual helpfulness. 

Just as here and there on the face of a woman sits the 
divine spirit of beauty, so that all cannot but love who 
look — so, presenting itself to us in all this mingled scene 
of air and ocean, plain and mountain, is a being of such 
loveliness that, did we but know with one accord i^ one 
stream, all our hearts would be carried in a perfect and 
willing service. It is not that we need to be made 
different ; we have but to look and gaze, and see that 
centre whereunto with joyful love aU created beings 
move. 

But not with eflbrtless wonder will our da)rs be filled, 
but in toil and strong exertion ; for, just as now we all 
labour and strive for an object, our service is bound up 
with things which we do— so then we find no rest from 
* labour, but the sense of solitude and isolation is gone. 
The bonds of brotherhood with our fellow-men grow 
strong, for we know one common purpose. And through 
the exquisite face of nature shines the spiritual light 
that gives us a great and never-failing comrade. 

Our task is a simple one— to lift from our mind that 
veil which somehow has fallen on us, to take that curious 
limitation from our perception, which at present is only 
transcended by inspiration. 

And the means to do it is by throwing aside our reason 
—by giving up the idea that what we think or are has 
any value We too often sit as judges of nature, wlien 
all we can be are her humble learners. We have but 
to drink in of the inexhaustible fulness of being, press- 
ing it close into our minds, and letting our pride of being 
able to foretell vanish into dust 

There is a curious passage in the works of Immanuel 
Kant,! in which he shows that space must be in the. 

* The Idea of tpsoe can **tticiit ans den Verliildiissen der 



So 



A Niw Era of Thought 



miod before we can observe things in space. Tor/' 
he 9SKf% ''since everything we conceive is conceived as 
being in spaoe^ there is nothing which conies before our 
minds from which the idea of spate can be derived ; 
it is equally present in the most rudimentary perception 
and' the most complete." Hence he says that space 
belongs to the perceiving soul itself. Without going 
into this argument to abstract regions, it has a great 
amount of practical truth. All our perceptions are of 
things in space ; we cannot think of any detail, how* 
ever limited or isolated, which is not in space. 

I^ence^ in order to exercise our perceptive powers. 
It is well to have prepared beforehand a strong appre- 
hension of space and space relations. 

And so, as we pass on, is it not easily conceivable 
tha^ with our power of higher space perception so 
rudimentary* and so unorganized, we should find it im- 
possible to perceive higher existences? That mode of 
perception which it belongs to us to exercise is wanting. 
What wonder, then, that we cannot see the objects 
which are ready, were but our own part done ? 

Think how much has come into human life through 
exercising the power of the three-dimensional space 
perception, and we can form some measure^ in a faint 
way, of what is in store for us. 

'. There is 9^ certain reluctance in us in bringing any- 
thing, which before has been a matter of feeling, within 
the domain of conscious reason. We do not like to 
explain why the grass is green, flowers bright, and, 
above all, why we have the feelings which we pass 
through. 

But this objection and instinctive reluctance is chiefly 

ansserm Ertcheimmg durch Erlahrung erborgt sein, sondem diete 
instere ErfiUmuif ist nur durch gedachte VorsteUung alierent 



PemplioH and Inspiration. 



8i 



derived from the fact that explaining has got to mean 
explaining away. We so often think that, a thing is 
explained, when it can be shown simply to be another 
form of something* which we know already. And, in 
fact, the wearied mind often does long to have a 
phenomenon shown to be merely a deduction from 
certain known laws. 

But explanation proper is not of this kind; it is 
introducing into the mind the new cojiteption which 
is indicated by the phenomenon ikeady present 
Nature consists of many entities towards the appre- 
hension of which we. strive. If for a time we break 
down the bounds which we have set up, and unify vast 
fields of observation under one common law, it is that 
the conceptions we formed at first are inadequate^ and 
must be replaced by greater ones. But it is always 
the case, that, to understand nature, a conception must 
be formed in the mind. This process of growth in the 
mental history is hidden ; but it is the really important 
one. The new conception satisfies more facts than the 
old ones, is truer phenomenally ; and the arguments for 
it are its simplicity, its power of accounting for many 
facts. But the conception has to be formed first And 
the real history of advance lies in the growth of the 
new conceptions which tvtty now and then come to 
light 

When the weather-wise savage looked at the sky at 
night, he saw many specks of yellow light, h'ke fire- 
flies^ sprinkled amidst whitish fleece; and sometimes 
the fleece remained, the fire-spots went, and rain came ; 
sometimes the fire-spots remained, and the night was 
fine. He did not see that the fire-points were ever the 
same, the clouds difierent; but by feeling dimly, he 
knew enough for his purpose. 

But when the thinking mind turned itself on these 

G 



83 



A Nm Era of Thought. 



appeanuices, there sprang upb«*-not all at once, but 
graduallyt— the knowledge of the sublime existences of 
the distant heavens, and all the lore of the marvellous 
forms of water, of air, and the movements of the earth. 
Surely these realities, in which lies a wealth of em- 
bodied poetry, are well worth the del^hted sensuous 
apprehension of the savage as he gazed. 

Perhaps something is lost, but in the realities, of 
which we know, there is compensation* And so, when 
we learn to understand the meaning of these mysterious 
changes, this course of natural events^ we shall find in 
the greater realities amongst which we move a fair 
exchange for the instinctive reverence, which they now 
awaken in us. 
V'~ In this book the task is taken up of forming the 
|d most simple and elementary of the great conceptions 
tha( are abput us. In the works of the poets, and still 
more in the pages of religious thinkers, lies an untold 
wealth of conception, the organisation of which in our 
eveiy-day intellectual life is the work <^ the practical 
educator. 

But none is capable of such simple demonstration 
and absolute presentation as this of higher space, and 
none so immediately opens our tyts to see the world 
as a different place. And, indeed, it is very instructive ; 
' for when the new conception is formed, it is found to 
be quite simple and natural We ask ourselves what 
we have gained; and we answer: Nothing; we have 
simply removed an obvious limitatioa 

And this is universally true ; it is not that we must 
rise to the higher by a long and laborious process. We 
may have a long and laborious process to go through, 
but, when we find the higher, it is this : we discover our 
true sdvei^, our essential being, the fact of our lives. 
In this cas4 we pass from the ridiculous limitation, to 



Pirception and Inspiration. 



83 



which our eyes and hands seem to be subject, of acting 
iii a mere section of space, to the fuller knowledge and 
feeling of space as it is. How do we pass to this truer 
intellectual life 7 Simply by observing, by laying aside 
our intellectual powers, and by looking at what is. 

We take that which is easiest to observe^ hot that 
which is easiest to define ; we Uke that which is the 
most definitely limited real thing, and use it as our 
touchstone whereby to explore nature. 

As it seems to me, Kant made the great and funda- r 
mental statement in philosophy when he exploded all .' 
previous systems, and all physics were reft from off the ^ 
perceiving souL But what he did once and for afl, was too ^ 
great to be a practical means of intellectual work. The ^ 
dynamic form of his absolute insight had to be found ; "" 
and it is in other works that the practical instances 
of the Kantian method are to be found For, instead 
of looking at the large foundations of knowledge, the 
ultimate principles of experience, late writers turned 
to the details of experience, and tested every pheno- 
menon, not with the question. What is this? but with 
the question, ** What makes me perceive thus 7 " 

And surely the question, as so put, is more capable 
of an answer; for it is only the percipient, as a subject 
of thought, about which we can speak. The absolute 
soul, since it is the thinker, can never be the subject of 
thought ; but, as physically conditioned, it can be thought 
about Thus we can never, without committing a 
ludicrous error, think of the mind of man except as 
a material organ of some kind ; and the path of dis« 
covery lies in investigating what the devious line of his 
thought history is due to, which winds between two 
domains of physics— the unknown conditions which - 
affect the percdver, the partially . known physics 
which constitute what we call the external worid. 



84 



A New Era of ThougtU. 



It is a pity to spend time over these reflections ; 
if they do not seem tame and poor compared to the 
practical apprehension which comes of working with 
the models^ then there is nothing in the whole subject 
ir In the little real objects which the reader has to 
handle and observe does not lie to him a poetry of a 
higher kind than any expiessed thought, then all these 
words are not only useless, but false. If, on the other 
hand, there Is true work to be done with them» then 
these suggestions will be felt to be but mean and 
insuflident apprehensions. 

For, in the simplest apprehensk>n of a higher space 
lies a knowledge of a reality which ls» to the realities 
we know, as spirit is to matter ; and yet to this new 
visfam all our solid facts and material conditions are 
but as a shadow Is to that which casts it In the 
awakening light of this new apprehension, the flimsy 
Wbrid quiveiB and shakes^ rigid solids flow and mingle, 
all our material limitations turn into graciousness, and 
the new fieU ^ possibility watU for us to look and 
behokL 



CHAPTER XL 

SPACE THE SCIENTIFIC BASIS OF ALTRUISM AND 
RELIGION. 

The reader will doubtless ask for some definite result 
corresponding to these words — ^something not of the 
nature of an h}rpothesis or a might-be. And in that I 
can only satisfy him after my own powers. My only 
strength is in detail and patience ; and if he will go 
through the practical part of the book, it will assuredly 
dawn upon him that here is the beginning of an answer 
to his request I only study the blocks and stones 
of the higher life But here they are definite enough. 
And the more eager he is for personal and spiritual truth, 
the more eagerly do I urge him to take up the practical 
work, for the true good comes to us through those who, 
aspiring greatly, still submit their aspirations to fact, 
and who, desiring to apprehend spirit; still are willing to 
manipulate matter. 

The particular problem at which I have worked forj 
more than ten years, has been completely solved. It 
possible for the mind to acquire a conception of higher\ 
space as adequate as that of our three-dimensional] 
space, and to use it in the same manner. 

There are two distinct ways of studying space— our 
familiar space at present in use. One is that of the 
analyst, who treats space relations by his algebra, and 
discovers marvellous relations. The other is that of the 
observer or mechanician, who studies the shapes of things 
In space directly. 

•s 



86 



A New Era of Thought. 



'A 



A practical designer of machines would not find the 
knowledge of geometrical analysis of immediate help to 
him ; and an artist or draughtsman still less sa 

Now, my inquiiy was, whether it was possible to get 
the same power of conception of four-dimensional space^ 
as the designer and draughtsman have of thrce-dimen- 
sional space. It is possible. 

And with this jpower it is possible for us to design 
machines in higher space^ and to conceive objects in 
this space, just as a diaughtsman or artist does. 

Analytical sldll is not of much use in designing a 
statue or inventing a machine, or in appreciating the 
detail cif either a work of art or a mechanical con- 
trivancc; 

And hitherto the study of four-dimensional space has 
been conducted by analysis. Here, for the first time, 
the fact of the power of conception of four-dimensional 
space is demonstrated, and the means of educating it 
are givea 

And I propose a complete system of work, of which 
the vdume on four space » is the first instalment 

I shall bring forward a complete system of four- 
dimensional thought— mechanics, science, and art The 
nccessaiy condition i^ that the mind acquire the power 
of using four-dimensional space as it now does three* 
dimensionaL 

And there is another condition which is no less im- 
portant We can never see, for insUnce, four-di- 
mensional pictures with our bodily eyes, but we can 
with our mental and inner eye. The condition is, that 
we should acquire the power of mentally canying a 
great number of details. 

I( for instance, we could think of the human body 



Space the Basis of Altruism and Rdigum. 87 

right down to every minute part in its right position, 
and conceive its aspect, we should have a four*di- 
mensional picture which is a solid structure. Now, to 
do this, we must form the habit of mental painting, that 
is, of putting definite colours in definite positions, not 
with our hands on paper, but with our minds in thought, 
so that we can recall, alter, and view complicated arrange* 
ments of colour existing in thought with the same ease 
with which we can paint 00 canvas. This is simply an 
afl*dir of industry ; and the mental power latent in us in 
this direction is simply marvellous. 

In any picture, a stroke of the brush put on without 
thought is valueless. The artist is not conscious of the 
thought process he goes through. For our purpose it 
is necessary that the manipulation of colour and form 
which the artbt goes through unconsciously, should be- 
come a conscious power, and that, at whatever sacrifice 
of immediate beauty, the art of mental painting should 
exist beside our more unconscious art All that I mean 
is this — ^that in the course of our campaign it is necessaiy 
to take up the task of learning pictures by heart, so 
that, just as an artist thinks over the outlines of a figure 
he wants to draw, so we think over each stroke in our 
pictures. The means by which this can be done will.be 
given in a future volume. 

We throw ourselves on an enteiprise in which we have 
. to leave altogether the direct presentation to the senses. 
We must acquire a sense-perception and memory of so 
keen and accurate a kind that we can build up mental 
pictures of greater complexity than any which we can 
see. We have a vast work of oiganixatioo, but it is 
merely oiganization. The power really exists and 
shows Itself when it is looked for. 

Much fault may be found with the ^system of organi- 
sation which I have adopted, but it is the survivor of 



88 



A New Era of Thought. 



many attempts ; and although I could better it in parts, 
still I think it b best to use it until, the full importance 
of the subject being realized, it will be the lifework of 
men of science to reorganize the methods. 

The one thing on which I must insist is this— that 
knowledge is of no value, it does not exist unless it 
comes into the mind. To know that a thing must be is 
no use at all. It must be clearly realized, and in detail 
as it is^ before it can be used. 

A whole world swims before u% the apprehension 
of which simply demands a patient cultivation of our 
powers ; and then, when the faculty is formed, we shall 
Recognize what the universe in which we are is like. We 
shall learn about ourselves and pass into a new domain. 
And I would speak to some minds who, like myself, 
share to a large extent the feeling of unsettledness and 
unfixedness of our present knowledge. 
« Religion has suflfered in some respects from the in-* 
accuraqr of its statements ; and it is not always seen 
that it consists of two parts— one a set of rules as to the 
management of our relations to the physical world about 
us, and to our own bodies ; another, a set of rules as to 
our relationship to beings higher than ourselves. 

Now, on the former of these subjects, on physical facts, 
on the laws of health, science has a fair standing ground 
of criticism, and can correct the religious doctrines in 
Ikiany important respects. 

But on the other part, of the subject matter, as to our 
relationship to beings higher than ourselves, science 
has not yet the materials for judging. The proposition 
which underlies this book is, that we should begin to 
acquire the faculties for judging. 

To judges we must first appreciate ; and how far we 
are from appreciating with science the fundamental 
rdigious doctrines I leave to any one to judge. 



Spaci the Basis of Altruism and Reltgion. 89 

There is absolutely no scientific basis for morality, 
using morality in the higher sense of other than a code 
of rules to promote the greatest physical and mental 
health and growth of a human being. Science does not 
give us any information which is not equally acceptable 
to the most selfish and most generous man ; it simply 
tells him of means by which he may attain his own 
ends, it does not show him ends. 

The prosecution of science is an ennobling pursuit ; 
but it is of scientific knowledge that I am now speaking 
in itself. We have no scientific knowledge of any exist- 
ences higher than ourselves- at least, not recognized 
as higher. But we have abundant knowledge of the 
actions of beings less developed than ourselves, from 
the striking unanimity with which all inorganic beings 
tend to move towards the earth's centre, to the almost 
equally uniform modes of response in elementary or- 
ganized matter to different stimuli. 

The question may be put : In what way do we come 
into contact with these higher beings at present ? And 
evidently the answer is, In those ways in which we 
tend to form organic unions — ^unions in which the activi- 
ties of individuals coalesce in a living way. 

The coherence of a military empire or of a subju- 
gated population, presenting no natural nucleus of 
growth, is not one through which we should hope 
to grow into direct contact with our higher destinies. 
But in friendship, in volunUry associations, and above 
all, in the family, we tend towards our greater life. 

And it seems that the instincts of women are much 
more relative to this, the most fundamental and import- 
* ant side of life, than are those of men. In fact; until 
we know, the line of advance had better be left to the 
feeling of women, as they organize the home and the 
social life spreading out therefrom. It is difficutt,perhap^ 



90 



A New Era of ThwghL 



(or a man to be still and perceive ; but if he is so, he 
finds that what, when thwarted, are meaningless caprices 
and empty emotionalities, are, on the part of woman, 
when allowed to grow freely and unchecked, the first 
b^nnings of a new life — ^the shadowy filaments, as it 
weie^ by which an organism begins to coagulate to* 
getber from the medium in which it makes its appearance. 

In veiy many respects men have to make the condi- 
tions^ and then learn to recognize. How can we see 
the higher beings about us, when we cannot even 
conceive the simplest higher shapes ? We may talk 
about spacer and use big words, but» after all, the prefer* 
^le way of putting our efibrts is this : let us look first 
at the simplest facts of higher existence, and then, when 
we have learnt to realize these, we shall be able to see 
what the world presents And then, also, light will be 
thrown on the constituent organisms of our own bodies, . 
When we see in the thorough development of our social 
life a relation between ourselves and a larger organism 
similar to that which exists be twe en us and the minute 
constituents of our frame. 

The problem, as it comes to me, is this : it is dearly 
demonstrated that self-regSLrd is to be put on one side— 
and self-regard in every respect — ^not only should things 
painful and arduous be done^ but things degrading and 
vilc^ so that they serve. 

. I am to sign any list of any number of deeds which 
the most foul imagination can suggest, as things which I 
would do did the occasion come when I could benefit 
another by doing them ; and, in fact, there is to be no 
characteristic in any action which I would shrink from 
did the .occasion come when it presented itself to be 
done for another's sake. And I believe that the soul 
it absolutely unstained by the action, provided the re* 
gaid U fot another. 



! 



I 



^ 



Space the Basis of Altruism and Religion. 91 

But this is, in truth, a dangerous doctrine ; at one 
sweep it puts away all absolute commandments, all 
absolute verdicts of right about things, and leaves the 
agent to his own judgment 

It is a kind of rule of life which requires most abso* 
lute openness, and demands that society should frame 
severe and insuperable regulations ; for otherwise, with 
the motives of the individual thus liberated from absolute 
law, endless varieties of conduct would spring forth, 
and the wisdom of individual men is hardly enough to 
justify their irresponsible action. 

Still, it does seem that, as an ideal, the absolute 
absence of self-regard is to be aimed at 

With a strong religious basis^ this would work no 
harm, for the rules of life, as laid down by religions, 
would suffice. But there are many who do not accept 
these rules as any absolute indication of the will of 
God, but only as the regulations of good men, which 
have a claim to respect and nothing more. 

And thus it seems to me that altruism— thorough* 
going altruism— hands over those who regard it as an 
ideal, and who are also of a sceptical turn of mind, to 
the most absolute unfixedness of theory, and,, very pos* 
sibly, to the greatest errors in life. 

And here we come to the point where the study of 
space becomes so important 

For if this rule of altruism is the right one, if it 
appeals with a great invitation to us, we need not there* 
fore try it with less precaution than we should use in 
other affairs of infinitely less importance. When we 
want to know if a plank will bear, we entrust it with 
a different load from that of a human body. 

And if this law of altruism is the true one, let us try 
it where failure will not mean the ruin of human 
beingsi 



9^ 



A New Era of Thought 



Novff in knowledge, pure altruism means so to buiy 
tbe mind in the thing known that all particular relations 
of one's self pass away. The altruistic knowledge of 
tbe heavens wouM be, to feel that the stars were vast 
bodies, and that I am moving rapidly. It would be, to 
know this, not as a matter of theory, but as a matter 
of habitual feeling. 

Whether this is possible, I do not know ; but a some- 
what simihur attempt can be made with much simpler 
means. 

In a different place I have described the process of 
acquiring an altruistic knowledge -of a block of cubes ; 
,and the results of the laborious processes involved are 
well worth the trouble. For as a clearly demonstrable 
fact this comes before one. To acquire an absolute 
knowledge of a block of cubes, so that all self relations 
are cast out, means that one has to take the view of a 
Jiigher being. 

It suddenly comes before one, that the particular re- 
lations which are so fixed and important, and seem so 
absolutely sure when one begins the process of learning, 
are by no means absolute facts, but marks of a singular 
limitation, almost a degradation, on one's own part In 
the determined attempt to know the most insignificant 
object perfectly and thoroughly, there flashes before 
one*s eyes an exbtence infinitely higher than one's own. 
And with that vision there comes, — I do not speak 
from my own experience only,— a conviction that our 
existence also is not what we suppose — that this 
bodily self of ours is but a limit toa And the question 
of altruism, as against self-r^ard, seems almost to 
vanish, for by altruism we come to know what we truly 



''What we truly are,*I do not mean apart from space 
and matter, but what we really are as beings having a 



i 



^< 



ii 



S/ace the Basis of AUrmsm and Rdigum. 93 

space existence ; for our way of thinking about existence 
is to conceive it as the relations of bodies in space. To 
think is to conceive realities in space. 

Just as, to explore the distant stars of the heavens, a 
particular material arrangement is necessary which we 
call a telescope, so to explore the nature of the beings 
who are higher than us, a mental arrangement is ne- 
cessary. We must prepare our power of thinking as 
we prepare a more extended power of looking. We 
want a structure developed inside the skull for the 
one purpose, while an exterior telescope will do for the 
other. 

And thus it seems that the diiliculties which we first 
apprehended fall away. 

To us, looking with half-blinded eyes at merely our 
own little slice of existence, our filmy all, it seemed 
that altruism meant disorder, vagary, danger. 

But when we put it into practice in knowledge, we 
find that it means the direct revelation of a higher 
being and a call to us to participate ourselves too in a 
higher life — ^nay, a consciousness comes that we are 
higher than we know. 

And so with our moral life as with our intellectual 
life. Is it not the case that those, who truly accept the 
rule of altruism, learn liTc in new dangerous ways ? 

It is true that we must give up the precepts of religion 
as being the will of God ; but then we shall learn that 
the will of God shows itself partly in the religious pre- 
cepts^ and comes to be more fully and more plainly 
known as an inward spirit 

And that difficulty, too, about what we may do and 
what we may not, vanishes also. For, if it b the same 
about our fdlow-creatures as it is about the block of 
cubes, when we have thrown out the self-rq[ard from 
our relationship to them, we shall feel towards them as 



94 



A New Era of Thought. 



a higher being than man feels towards them, ive shall 
fed towards them as they are in their true selves, not in 
their outward forms, but as eternal loving spirits. 

And then those instincts which humanityr feels with a 
secret impulse to be sacred and higher than any tem- 
poraiy good will be justified — or fulfilled 

There are two tendendes^-one towards the direct 
cultivation of the religious perceptionsi the other to re-^ 
dudng everything to reason. It will be but just for the * 
exponents of the latter tendency to look at the whole 
universe^ not the. mere section of it which we know, be- 
fore they deal authoritatively with the higher parts of 
religion. 

And those who fed the immanence of a higher life in 
us will be needed in this outlook on the wider field of 
reality, so that they, being fitted to recognize, may tell 
us what lies ready for us to know» 

The true path of wisdom consists in sedng that our 
intdlect is foolishness— that our conclusions are absurd 
and mistaken, not in Speculating on the world as a form 
of thought projected fi-om the thinking prindple within 
OS — rather to be amazed that our thought has so limited 
the world and hidden from us its real existences To 
think of oursdves as any other than things in space and 
subject to material conditions, is absurd, it is absurd on 
dther. of two hypotheses. If we are really things in 
spacer then of coune it is absurd to think of oursdves 
as if we were not sa Qn the other hand, if we are not 
things in q>ace^ then conceiving in space is the mode 
in which that unknown which we afo exists as a mind. 
Its mentd action is space-conception, and then to give 
up the idea of oursdves as in spacer is not to get a truer 
idea, but to lose the only power of apprdienskm of our« 
sdvts which we possess 

And yeC th^ i% it most be confessed, one way hi 






Space the Basis of Altruism and Religion. 95 

which it may be possible for us to think without think- 
ing of things in space. 

That way is, not to abandon the use of space-thought» 
but to pass through it 

When we think of space, we have to think of It as in- 
finity extended, and we have to think of it as of infinite 
dimensions. Now, as I have shown in • The Law of the 
Valley," * when we come upon infinity in any mode of 
our thought, it b a sign that that mode of thought is 
dealing with a higher reality than it is adapted for, and 
in struggling to represent it, can only do so by an in- 
finite number of terms. Now, space has an infinite 
number of positions and turns, and this may be due to 
the attempt forced upon us to think of things higher 
than space as in space. If so, then the way to get rid 
of space from our thoughts, is, not to go away from it, 
but to pass through it— to think about larger and larger 
systems of space, and space of more and more dimen- 
sions, till at last we get to such a represenUtion in 
space of what is higher than space, that we can pass 
from the space-thought to the more absolute thought 
without that leap which would be necessary if we were 
to try to pass beyond space with our present very in- 
adequate representation in it of what really is. 

Again and again has human nature aspired and 
fallen. The vision has presented itself of a law which 
was love, a duty which carried away the enthusiasm, 
and in which the conflict of the higherand lower natures 
ceased because all was enlisted in one loving service 
But again and again have such attempts failed. The 
common-sense view, that man Is subject to law, external 
law, remains— that there are fates whom he must pro- 
pitiate and obey. And there is a strong sharp curb, 

> " Sdeuce Romances,* No. IL 



96 



A New Era of Thought 



which, if It be not brought to bear by the will, is soon 
pulled t^ht by the world, and one more tragedy is 
enacted, and the over-confident soul is brought low. 

And the rock on which such attempts alwa]^ split, 
is in tlie indulgence of some limited passion. Some 
one object fills the soul with its image, and in devotion 
to that, other things are sacrificed, until at last all 
comes to ruin* 

But what do^ this mean P* Surely it b simply this, 
that where there should be knowledge there b ignorance. 
It b not that there b too much devotion, too much 
passion, but that we are ignorant and blind, and 
Wiuider in error We do not know what it is we care 
for, and waste our effort on the appearance. There b 
no such 'thing as wrong love ; there b good love and 
bad knowledge, and men who err, clasp phantoms to 
themselves. Religfon is but the search for realities ; 
and thought; conscious of its own limitations^ b its 
bestakL 

Let a man care for any one object — let hb r^ard 
for it be as concentrated and exclusive as you will, 
there will be no danger if he truly apprehends that 
which he cares for. Its true being b bound up with 
aU the rest of exbtencc; and, if hb regard b true to 
one; then, if tfiat one b really known, hb regard is 
truetoalL 

There b a question sometimes asked, which shows 
the mere formalbm into which we have fallen. 

Weask: Whatbtheendofexbtence? A mere play 
on words I For to conceive exbtence b to fed ends. 
The knowledge of existence b the caring ibr objects, 
tile fear of dangerii the anxieties of love» Immersed 
in theses the triviali^of the question, what b the end of 
existence ? becomes obvious. If, however, letting reality 
fiMle awa/i we phy with wofds» some questions of thb 



Spau the Basis of Altruism and Religion. 97 

kind are possible ; but they are mere questions of words, 
and all content and meaning has passed out of them. 

The task before us b thb : we strive to find out that 
physical unity, that body which men are parts of, and 
in the life of which their true unity lies. The exbtence 
of thb one body we know from the utterances of those 
whom we cannot but fed to be inspired ; we fed certain 
tendencies in ourselves which cannot be explained 
except by a supposition of thb kind. 

And, now, we set to work deliberatdy to form in 
our minds the means of investigation, the faculty of 
higher-space conception. To our ordinaiy space- 
thought, men are isolated, dbtinct, in great measure 
antagonbtic. But with the first use of the weapon of 
higher thought, it b easily seen that all men may really 
be members of one body, thdr isolation may be but an 
affair of limited consdousness. There is, of course; no 
value as science in such a supposition. But it suggests 
to us many possibilities ; it reveals to us the confined 
nature of our present physical views, and stimulates us 
to undertake the work necessary to enable us to deal 
adequately with the subject 

The work b entirely practical and detailed ; it b the 
elaboration, beginning from the simplest objects of an 
experience in thought, of a higher-space world. 

To begin it, we take up those detaib of position and 
relation which are generally relegated to symbolism or 
unconsdous apprehension, and bring these waste pro- 
ducts of thought into the central position of the labora- 
toiy of the mind. We turn all our attention on the 
most simple and obvious detaib of our eveiy-day ex- 
perience, and thence we build up a conception of the 
fundamental facts of position and arrangement in a 
higher world. We next study more complicated h^her 
shapes, and get our space perception drilled and dis- 

H 



98 



Nm Era of Tkattgkt. 



dplined. Tben we proceed to put ft content into our 
fitamevrork 

The means of doing this are tw^old— observfttion 
and insfrfradon. 

As to observation, it b hardly possible to describe 
the feelings of that investigator who shall distinctly 
trace in the physical world, and experimentally de- 
monstrate the existence of the higher-space facts which 
are so curiously hidden from us. He will lay the first 
stone for the observation and knowledge of the higher 
beings to whom we are related. 

As to the other means, it is obvious, surely, that if 
tiiere has ever been inspiration, there. is inspiration 
now. Inspiration is not a unique phenomenon. It has 
existed in absolutely marvellous degree in some o£ the 
teachers of the ancient world ; but that, whatever it 
waS| which thqr possessed, must be present now, and, 
if we could isolate it, be a demonstrable fact 

And I would propose to define inspiration as the 
iaculty, which, to take a particular instance, does the 
followii^:*-* 

If a square penetrates a line comerwise^ it marks 
out on the line a s^ment bounded by two points— that 
i% we suppose a line drawn on a piece of paper, and 
a square lybig on the paper to be pushed so that its 
oomer passes over the line. Then, supposing the paper 
' and the li^e to be in the same plane^ the line b inter- 
rapted by the square; and, of the square^ all that b 
observable in the line^ b a s^ment bounded by two 
po in ts^ 

Next; suppose a cube to be pushed comerwise 
throiq^h a ^ane^ and let the plane make a section of 
the cube The section will be a plane figure^ and it 
will be a triangle; 

Now, first, the sectkin of a square by a line b a 



Space the Basis of Altruism and Religion. 99 

segment bounded by two points ; second, the section 
of a cube by a plane b a triangle bounded by three 
lines. 

Hence^ we infer that the section of a figure in four 
dimensions analogous to a cube, by three-dimensional 
spacer will be a tetrahedron— a figure bounded by four, 
planes. 

Thb b found to be true ; with a little familiarity 
with four-dimensional movements thb b seen to be 
obvious. But I would define inspiration as the faculty 
by which without actual experience thb conclusion 
b formed. 

How it b we come to this conclusion I am perfectly 
unable to say. Somehow, looking at mere formal con- 
siderations, there comes into the mind a conclusion 
about something beyond the range of actual experience. 

We may call thb reasoning from analogy ; but using 
thb phrase does not explain the process. It seems to 
me just as rational to say that the facts of the line and 
plane remind us of facts which we know already about 
four-dimensional figures— that they tend to bring these 
facts out into consciousness, as Plato shows with the 
bo/s knowledge of the cube. We must be really four- 
dimensional creatures^ or we could not thmk about four 
dimensions. 

But whatever name we give to thb peculiar and in- 
explicable faculty, that we do possess it b certain ; and 
in our investigations it will be of service to us. We 
must carefully investigate existence in a plane world, 
and then, making sure, and impres^ng on our inward 
sens^ as we go, every step we take with ttg^sA to a 
higher world, 4re shall be reminded continually of fresh 
possibilities of our higher exbtence. 



PART 11. 



CHAPTER I. 

THREE-SPACE. GENESIS OF A CUBE. APPEARANCES 
OF A CUBS TO A PLANE-BEING. 

The models consist of a set of eight and a set of four 
cubes. They are marked with different colours^ so as 
to show the properties of the figure in H^her Space, to 
which they belong. 

The simplest figure in one-dimensional space, that is, 
in a straight line, b a straight line bounded at the two 
extremities. The figure in this case consists of a length 
bounded by two points. 

Looking at Cube i, and placing it so that the figure i 
is uppermost, we notice a straight line in contact with 
the table, which is coloured Orange. It b^ns in a 
Gold point and ends in a Fawn point The Orange 
extends to some distance on two faces of the Cube ; but 
for our present purpose we suppose it to be simply a 
thin line. 

This line we conceive to be generated in the following 
way. Let a point move and trace out a line. Let the 
point be the Gold point, and let it, moving, trace out the 
Orange line and terminate in the Fawn point Thus 
the figure <;onsists of the. point at which it begins^ the 
point at which it ends, and the portion between. We 
may suppose the point to start as a Gold point, to 



I02 



Niw Era of Thought. 



change its colour to Orange during the motion, and 
when it stops to become Fawn. The motion we suppose 
finom left to right, and its direction we call X. 

If, now, this Orange line move away from us at right 
angles^ it will trace out a square. Let this be the Black 
square^ which is seen underneath Model I. The points, 
which bound the line^ will during this motion trace out 
Itnes^ and to these lines there will be terminal points. 
AIso^ the Square will be terminated by a line on the. 
opposite side. Let the Gold point in moving away 
trace oat a Blue line and end in a Buff point ; the Fawn 
point a Crimson line ending In a Terracotta point 
Tlfe Orange line, having traced a Blade square^ ends in 
a Gieen-grey line. This direction, away from the 
observer, we caU Y. 

Now, let the whole Black square traced out by the 
Orange line move upwards at right angles. It will 
trace out a new figure^ a Cube. And the edges of the 
square^ while moving upwards, will trace out squares. 
Bounding the cube, and opposite to the Black square, 
will be another square. Let the Orange line moving 
upwards trace a Dark Blue square and end in a Reddish 
lue The Gold point traces a Brown line; the Fawn point 
trsMses aVrench-grey line^ and these lines end in a Light- 
blue and a Dull-purple point Let the Blue Une trace a 
• Vermilion square and end in a Deep-yellow line. Let 
the Buff point trace a Green line^ and end in a Red 
point The Green-grqr line* traces a Light-yellow 
square and ends in a Leaden line ; the Terracotta point 
tnces a Dark-slate line and ends in a Deep-blue point 
The Crimson Une traces a Blue-green square and ends 
fa a Br^t-Uue line. 

Finally, the Bhck square traces a Cube^ the colour of 
iriuch b invisible^ and ends in a white square. We 
suppose the colour of the cube to be a Light-buff The 



I 
ThreO'Spaa. Gmesis of a Cubo. 103 

upward direction we call Z. Thus we say : The Gold 
point moved Z| traces a Brown line^ and ends in a Light- 
blue point 

We can now clearly realise and reier to each region 
of the cube by a colour. 

At the Gold point, lines from three directions meet, 
the X line Orange, the Y line Blue, the Z line Brown. 

Thus we began with a figure of one dimension, a line, 
we passed on to a figure of two dimensions, a square, 
and .ended with a figure of three dimension^ a cube 



1 



The square represents a figure in two dimensions ; but 
if we want to realize what it is to a being in two 
dimensions, we must not look down on it Such a view 
could not be taken by a plane-being. 

Let us suppose a being moving on the surface of the 
Uble and unable to rise from it Let it not know that 
it is upon anything; but let it believe that the two 
directions and compounds of those two directions are all 
possible directions. Moreover, let it not ask the ques- 
tion : "* On what am I supported ?** Let it see no reason 
for any such question, but simply call the smooth surface, 
along which it moves, Space. 

Such a being could not tell the colour of the square 
traced by the Orange line. The square would be 
bounded by the lines which surround it, and only by 
breaking through one of those lines could the plane- 
being discover the colour of the square. 

In trying to realize the experience of a plane-being 
it is best to suppose that its two dimensions are upwards 
and sideways, U^ Z and X, because^ if there be any 
matter in the plane-world, it will, like matter in the 
solid world, exert attractions and repulsions. The 
matter, like the beings, must be supposed very thin, that 



I04 



N€w Era of Thought. 



i% of so sligfat thidcoess that it is quite unnoticed by the 
being, Now, if there be a vay large mass of such 
matter lying on the table, and a plane*being be free 
to move about it, he will be attracted to it in eveiy 
direction. ''Towards this huge mass'' would be 
"Down,** and « Away from if* would be ''Up/' just as 
"Towards the earth'' b to soUd beings "Down," and 
"Away from it" is "Up," at whatever part of the globe 
they may be Henc^ if we want to realize a plane* 
being's feelings we must keep the sense of up and down. 
Therefore we must use the Z direction, and it is more 
convenient to take Z and X than Z and Y. 
« Any direction lying between these is said to be com- 
pounded of the two ; for, if we move slantwise for some 
distance^ the point reached might have been also reached 
by going a certain distance X, and then a certain 
distance Z» or t^wriA 

•Let us- suppose the Orange line has moved Z, and 
traced the Dark-blue square ending in the Reddish line. 
If we now place a piece of stiff paper against the Dark- 
Uue square^ and suppose the plane*beings to move to 
and fro on that surface of the paper, which touches 
the square we shall have means of representing their 
experience. 

To obtain a more consistent view of their existence^ 
let us suppose the piece of paper extended, so that it 
cuts through our earth and comes out at the antipodes, 
fhiis cutting the earth in twa Then suppose all the 
earth removed away, both hemispheres vanishing, and 
only a veiy thin layer of matter left upon the paper on 
that side which touches the Dark-blue square. This 
represents what the world would be to a plane-being. 

It is of some importance to get the notion of the 
directions in a plane-world, as great difficulty arises 
Crmq our notkms of up and down. We miss the right 



Thru-Space. Ginesis 4>f a Cube. 105 

analogy if we conceive of a plane-world without the 
conception of up and down. 

A good plan is, to use a slanting surface, a stiff card 
or book cover, so placed that it slopes upwards to the 
eye. Then gravity acts as two forces. It acts (i) as a 
force pressing all particles upon the slanting surface into 
it, and (2) as a force of gravity along the plane^ making 
particles tend to slip down its incline. We may suppose 
tliat in a plane-world there are two such forces, one 
keeping the beings thereon to the plane, the other 
acting between bodies in it, and of such a nature that by 
virtue of it any large mass of plane-matter produces on 
small particles around it the same effects as the large 
mass of solid matter called our earth produces on small 
objects like our bodies situated around it In both cases 
the larger draws the smaller to itself, and creates the 
sensations of up and down. 

If we hold the cube so that its Dark-blue side touches 
a sheet of paper held upwards to the eye, and if we 
then look straight down along the paper, confining our 
view to that which is in actual contact with the paper, 
we see the same view of the cube as a plane-being 
would get We see a Light-blue point, a Reddish Un^ 
and a Dull-purple point The plane-being only sees a 
line, just as we only see a square of the cube. 

The line where the paper rests on the Uble may be 
taken as representative of the surface of the pUne- 
being's earth. It would be merely a line to him, but it 
would have the same property in relation to the plane- 
world, as a square has in relation to a solid world ; in 
neither case can the notion of what in the latter is 
termed solidity be quite excluded. If the plane-being 
broke through the line boundhig his earth, he would find 
more nuitter beyond it 

Let us now leave out of consideration the question of 



106 



New Era of Thought. 



'*up and down" In it plane-world. Let os no longer 
consider it in the vertical, or ZX, position, bnt simply 
take the surface (XY) of the Uble as that which sup- 
ports a plane-world Let us represent its inhabitants 
by thin pieces of paper, which are free to move over the 
surface of the tables but cannot rise from it Also, let 
the tiiickness (U^ height above the surface) of these 
beings be so small that thqr cannot discern it Lastly 
let us premise there is no attraction in their world, so 
that thqr have not any up and down. 

Placing Cube i in front of us, let us now ask how a 
plane-being could apprehend such a cube. The Black 
bee he could easily study. He would find it bounded 
by Gold point. Orange line. Fawn point, Crimson line^ 
and so on. And he would discover it was Black by 
cutting through any of these lines and entering it. 
(This operation would be equivalent to the mining of a 
solid being). 

But of what came above the Black square he would 
be completely ignorant Let us now suppose a square 
hole to be made in the table, so that the cube could 
pass through, and let the cube fit the opening so 
exactly that no trace of the cutting of the table be 
visible to the plane-being. If the cube began to pass 
through, it wodld seem to him simply to change, for of 
its motion he could not be aware, as he would not know 
the directk>n in which it moved. Let it pass down till 
the White square be just on a level with the surface of 
the taUe, The plane-being would then perceive a 
Light-blue point; a Reddish line, a Dull-purple point, a 
Bright-blue line^ and so on. These would surround a 
White square, which belonged to the same body as that 
to which the Black square belonged. But in thb body 
tiiere would be a dimension, which was not in the 
square. Our upward direction would not be appre- 



Threi-Spdce. Genesis of a Cube. 107 

bended by him directly. Motion from above down- 
wards would only be apprehended as a change in the 
figure before him. He would not say that he had before 
him different sections of a cube, but only a changing 
square. If he wanted to look at the upper square^ he- 
could only do so when the Black square had gone an 
inch below his plane. To study the upper square 
simultaneously with the lower, he would have to make 
a model of it, and then he could place it beside the 
lower one. 

Looking at the cube, we see that the Reddish line 
corresponds precisely to the Orange line, and the Deep- 
yellow to the Blue line. But if the plane-being had a 
model of theupper square, and placed it on the right- 
hand side of the Black square^ the Deep-yellow line 
would come next to the Crimson line of the Black 
square. There would be a discontinuity about it All 
that he could do would be to observe which part in the 
one square corresponded to which part in the other. 
Obviously too there lies something between the Black 
square and the White. 

Tlie plane-being would notice that when a line moves 
in a direction hot its own, it traces out a square. When 
the Orange line is moved away, it traces out the Black 
square. Tlie conception of a new direction thus ob* 
Uined, he would understand that the Orange line 
moving so would trace out a square, and the Blue line 
moving so would do the same. To us these squares 
are visible as wholes, the Dark-blue^ and the Vermilion. 
To him they would be matters of verbal definition 
rather than ascertained facts. However, given that he 
had the experience of a cube being pushed through his 
plane, he would know there was some figure, whereof 
his square was part, which was bounded by his square 
on one side^ and by a White square on another side. 



io8 



Nm Era of Thought. 



We have supposed him to make roodeb of these bouo* 
daries» a Black square and a White square. The Black 
square^ which is his solid matter, is only one boimdaiy 
of a figure in Higher Space. 

But we can suppose the cube to be presented to him 
otherwise than by passing through his plane. It can be 
turned round the Orange Une^ in which case the Blue 
line goes out, and, after a time^ the Brown line comes 
in. It must l^ noticed that the Brown line comes into 
a dfaectkm opposite to that in which the Blue line 
raa These two lines are at right angles to each other, 
and, if one be moved upwards till it b at right angles to 
the surface of the tables the other comes on to the sur- 
* faoe^ but runs in a direction opposite to that in which 
the first ran. Thus^ by turning the cube about the 
Orange line and the Blue line, different sides of it can 
be shown to a plane-being. By combining the two 
^processes of turning and pushing through the plane^ all 
the sides can be shown to the plane-being. For in- 
stance^ if the cube be turned so that the Dark-blue 
square be on the plane^ and it be then passed through, 
the Light-yellow square will come in. 

Now, if the plane-being made a set of models of 
these different appearances and studied them, he could 
ibrm some ratk>nal idea of the Higher Solid which 
produced them. He would become able to give some 
consistent account of the properties of this new kind 
of existence; he could say what came bto his plane 
spacer if the other space penetrated the phme edge-wise 
or comer-wise, and could describe all that would come 
In as it turned about in any way. 

He wouM have six models. Let us condder two of 
them— the Black and the White squares. We can ob- 
serve them oo the dibe. Every colour on the one is 
dtfefent.fiom every ccdour on the other. If w 









Three-Space. Genesis of a Cube. 109 

ask what lies between the Orange line and the Reddish 
line^ we know it is a square, for the Orange line moving 
in any direction gives a square. And, if the six models 
were before the plane-being, he could easily select that 
which showed what he wanted. For that which lies, 
between Orange line and Reddish line must be bounded 
by Orange and Reddish lines. He would search among 
the six models lying beside each other on his plane, till 
he found the Dark-blue square. It is evident that only 
one other square differs in all its colours from the Black 
square^ viz., the White square. For it is entirely sepa^ 
rate. The others meet it in one of their lines. Tliis 
total difference exists in all the pairs of opposite sur- 
faces on the cube. 

Now, suppose the plane-being asked himself what 
would appear if the cube turned round the Blue line. 
The cube would begin to pass through his space. The 
Crimson line would disappear beneath the plane and 
the Blue-green square would cut it, so that opposite to 
the Blue line in the plane there would be a Blue-green 
line. The French-grey line and the Dark-slate line 
would be cut in points, and from the Gold point to the 
French-grey point would be a Dark-blue line; and 
opposite to it would be a Light-yellow line, from the 
Buff point to the Dark-slate point Thus the figure in 
the plane world would be an oblong instead of a square^ 
and the interior of it would be of the same Light-buff 
colour as the interior of the cube. It b assumed that 
the plane closes up round the passing cube^ as the sur- 
face of a liquid does round any object immersed 

But, in order to apprehend what would take place 
when thb twbting round the Blue line bq;an, the plane* 
being would halve to set to woric by parts. He has no 
conception of what a solid would do hi twistbg, but 
he knows what a plane does. Let him, dien, instead 



no 



New Era of ThugfU. 



of thinking of the whole Black square^ think only of 
the Orange line. The Dark-blue square stands on it 
As far as this square is concerned, twisting round the 
Blue line ts the same as twisting round the Gold point 
Let him imagine himself in that plane at right angles to 
his plane-world, which contains the Dark-blue square. 
Let him keep his attention fixed on the line where the 
two planes meet^ viz., that which is at first marked by 
the Orange line. We will call this line the line of his 
plane^ for all that he knows of his own plane is this 
line. Now, let the Dark-blue square turn round the 
Gold point. The Orange line at once dips below 
the line of his plane, and the Dark-blue square passes 
' through it Therefore, in his plane he will see a 
Park-blue line in place of the Orange one. And in 
place of the Fawn point, only further off from the Gold 
point, will be a French-grey point The Diagrams 
(i), (2) show how the cube appears as it is before and 
* after the turning. G is the Gold, F the Fawn point 
In (2) G is unmoved, and the plane is cut by the French- 
grey line^ Gr. 

Instead of imagining a direction he did not know, the 
plane- being could think of the Dark-blue square as 
l}dng in his plane. But in this case the Black square 
would be out off his plane^ and only the Orange line 
'would remain in it Diagram (3) shows the Dark-blue 
square lying in his planer and Diagram (4) shows it 
tomingYound the Gold point Here, instead of think- 
ing about his plane and also that at right angles to it; 
he has only to think how the square turning round the 
Gdd point will cut the line^ which runs left to right 
fitMB G, vis., the dotted line. The French-grey line is 
cut by the dotted line in a point To find out what 
would €ome in at other parts, he need only treat a 
nnmber of the plane sections of the cube perpendicular 




\j [^ 



F.,.r 



r.r3. 




a'u 



F^S. 



[TV iter A tM. 



Thret'Spac*. Genesis of a Cube* 1 1 1 



i 

I 

I 

i; 

i 






to the Black dquare in the same manner as he had 
treated the Dark-blue square. Every such section would 
turn round a point, as the whole cube turned round the 
Blue line, llius he would treat the cube as a number 
of squares by taking parallel sections from the Dark- 
blue to the Light-yellow square, and he would turn 
each of these round a comer of the same colour as the 
Blue line. Combining these series of appearances, he 
would discover what came into his plane as the cube 
turned round the Blue line. Thus, the problem of the 
turning of the cube could be settled by the consideration 
of the turnings of a number of squares. 

As the cube turned, a number of different appear- 
ances would be presented to the plane -being. The 
Black square would change into a Light-buft* oUong, 
with Dark-blue, Blue-green, Light-yellow, and Blue 
sidfs, and would gradually elongate itself until it be- 
came as long as the diagonal of the square side of 
the cube; and then the bounding line opposite to the 
Blue line would change from Blue-green to Bright-blue^ 
the other lines remaining the same colour. If the cube 
then turned still further, the Bright-blue line would 
become White, and the oblong would diminish in length. 
It would in time become a Vermilion ^uare^ with a 
Deep-yellow line opposite to the Blue line. It would 
then pass wholly below the plane^ and only the Blue line 
would remaia 

If the turning were continued till half a revolution 
had been accomplished, tbe Black square would come 
in again. But now it would come up into the plane 
from underneath. It would appear as a Black square 
exactly similar to the first ; but the Orange line^ in- 
stead of running left to right from Gold point, would 
run right to left. The square would be the same^ only 
differently disposed with regard to the Blue line It 



113 



New Era of Thought. 



would be die looking-glass Image of the first square. 
There would be a difference in respect of the lie of the 
partides of which It was composed. If the plane-being 
eoold examine its thickness, he would find that particles 
whicfap in the first case, lay above others, now lay below 
them. But, if he were really a plane-being, he would 
have no idea of thickness in his squares, and he would 
find them botfi quite identicaL Only the one would be 
to the odier as if it had been pulled through itseUl 
In this phenomenon of symmetry he would apprehend 
the difference of fhe.lie of the line, which went in the» 
to Unif unknown direction of up-and-down* 



\ 



CHAPTER II. 

FURTHER APPEARANCES OF A CUBE TO A 
PLANE-BEING. 

Before leaving the observation of the cube, it is well 
to look at it for a moment as it would appear to a 
plane- being, in whose world there was such a fact 
as attraction. To do this, let the cube rest on the table, 
so that its Dark-blue face is perpendicular in front of 
us. Now, let a sheet of paper be placed in contact with 
the Dark-blue square. Let up and sideways be the 
two dimensions of the plane-being, and away the un- 
known direction. Let the line where the paper meets 
the table, represent the surface of his earth. Then, 
there is to him, as all that he can apprehend of the 
cube, a Dark-blue square standing upright ; and, when 
we look over the edge of the paper, and regard merely 
the part in contact with the paper, we see what the 
plane-being would see. 

If the cube be turned round the up line, the Brown 
line, the Orange line will pass to the near side of 
the paper, and the section made by the cube in the 
paper will be an oblong. Such an oblong can be 
cut out ; and when the cube is fitted into it, it can 
be seen that it is bounded by a Brown line and a 
Blue-green line opposite thereto, while the other boun- 
daries are Black and White lines. Next, if we take 
a section half-way between the Black and WhiU 

««i I 



114 New Era of Thought. 

aqoaresi we shall have a square cutting the plane of 
the aforesaid paper in a single line. With regard to 
this section, all we have to inquire iS| What will Uke 
the place of this line as the cube turns ? Obviously, the 
line will elongate From a Dark-blue line it will change 
to a Laght-buff line^ the colour of the inside of tlie 
section, and will terminate in a Blue-green point instead 
of a.French*grey. Again, it is obvious that, if the cube 
turns round the Orange linc^ it wiU give rise to a scries 
of oUoogs, stretching upwards. This turning can be 
continued till the cpbe is wholly on the near side of the 
paper, and only the Orange line remains And, when 
the^cobe has made half a revolution, the Dark-blue 
square will return into the plane ; but it will run down- 
wards instead of upwards as at first Thereafter, if 
the cube turn furtiier, a series of oblongs will appear, 
all running downwards from the Orange line. Hcnce^ 
if all the appearances produced by the revolution of the 
cube have to be shown, it must be supposed to be raised 
Home distance above the plane-being's earth, so that 
those appearances may be shown which occur when it 
is turned round the Orange line downwards, as well 
as when it is turned upwards. The unknown direction 
comes into the plane citiier upwards or downwards, but 
there is no special connection between it and cither 
gA these directions. If it come in upwards, the Brown 
line goes nearwards or - Y ; if it come in downwards, 
or- Z» the Brown line goes away, or Y. 

Let us consider more closely the directions which the 
plane-beii« would have Firstiy, he would have up-and- 
down, that is, away from his eartii and towards it on 
the plane of the paper, the surface of his earth being 
the line where the paper meeU the Uble Then, if he 
moved along the surface of his eartii» tiiere would only 
bealiaefor himtomove in, the Une runm'ng right and 



Appearances of a Cube to a Plane-Being. 115 

left. But, being the direction of his movement, he 
would say it ran forwards and backwards. Thus he 
would simply have the words up and down, forwards 
and backwards, and the expressions right and left would 
have no meaning for him. If he were to frame a notion 
of a world in higher dimensions, he must invent new 
words for distinctions not within his experience. 

To repeat the observations already made, let the cube 
be held in front of the observer, and suppose the Dark- 
blue square extended on every side so as to form a 
plane. Then let this plane be considered as independent 
of the Dark-blue square. Now, holding the Brown line 
between finger and thumb, and touching its extremities, 
the Gold and Light-blue points, turn the cube round the 
Brown line The Dark-blue square will leave the plane, 
the Orange line will tend towards the -^Y direction, and 
the Blue line will finally come into the plane pointing 
in the +X direction. If we move the cube so that the 
line which leaves the plane runs -hY, then the line 
which before ran +Y will come into the plane in the 
direction opposite to that of the line which has left the 
plane The Blue line, which runs in the unknown direc* 
tion can come into either of the two known directions of 
the plane It can take the place of the Orange line 
by turning the cube round the Brown line, or the place 
of the Brown line by turning it round the Orange line 
If the plane-being made models to represent these two 
appearances of the cube, he would have identically the 
same line, the Blue line, running in one of his known 
directions in the first model, and in the other of his 
known directions in the second. In studying the cube 
he would find it best to turn it so that the line of un- 
known direction ran in that direction in the positive 
sense In that case, it would come into the plane in 
the negative sense of the known directions. 



Ii6 



New Era of Thought 



Starting with the cube in front of the observer, there 
are two ways in which the Vermilion square can be 
brought into the ims^inary plane; that is the extension 
of the Dark-blue square. If the cube turn round the 
Brown line so that the Orange line goes away, (i>. + Y), 
the Vermilion square comes in on the left of the Brown 
line; If it turn in the opposite direction, the Vermih'on 
square comes in on the right of the Brown line. Thus, 
if we identify the plane-being with the Brown line, the 
Vermilion square would appear either behind or before 
him. These two appearances of the Vermilion square 
would seem identical, but they could not be made to 
eoinctde by any movement in the plane. The diagram 
(Fig. 5.) shows the diiference in them. It is obvious that 
no turn in the plane could put one in the place of the 
other, part for part Thus the plane-being apprehends 
the reversal of the unknown direction by the disposition 
of his figures. If a figure, which lay on one side of a line, 
changed into an identical figure on the other side of it, 
he could be sure that a 'line of the figure, which at first 
ran in the positive unknown direction, now ran in the 
negative unknown direction. 

We have dwelt at great length on the appearances, 

which a cube would'present to a plane-being, and it will 

be found that all the points which would be likely to 

•cause difficulty hereafter, have been explained in this 

obvious cascT. 

There is^ however, one other way, open to a plane- 
beii^ of studsring a cube; to which we must attend. 
TUs is, by st^y motion. Let the cube come into the 
im^^ary plane; which is the extension of the Dark- 
blue square, ij^ let it touch the piece of paper which 
is standing vertical on the table. Then let it travel 
tlvoii(^ this {dane at i^ht angles to it at the rate of an 
indi a minute; .The plane*being would first perceive 



Appearances of a Cube to a Plane- Being. 1 1 7 

a Dark-blue square, that is, he would see the coloured 
lines bounding that square, and enclosed therein would 
be what he would call a Dark-blue solids In the move- 
ment of the ctibe, however, this Dark-blue square would 
not last for more than a flash of time. (The edges and 
points on the models are made very large ; in reality 
they must be supposed very minute.) This Dark-blue 
square would be succeeded by one of the colour of the 
cube's interior, U. by a Light-buff square. But this 
colour of the interior would not be visible to the plane- 
being. He would go round the square on his plane, and 
would see the bounding lines, via. Vermilion, White, 
Blue-green, Black. And at the comers he would see 
Deep-yellow, Bright-blue, Crimson, and Blue points. 
These lines and points would really be those parts of 
the faces and lines of the cube, which were on the point 
of passing through his plane. Now, there would be one 
difference between the Dark-blue square and the Light- 
buff with their respective boundaries. The first only 
lasted for a flash ; the second would last for a minute or 
all but a minute. Consider the Vermilion square. It 
appears to the plane-being as a line. The Brown line 
also appears to him as a line. But there is a difference 
between them. The Brown line only lasts for a flash, 
whereas the Vermilion line lasts jfor a minute. Hence, 
in this mode of presentation, we may say that for a 
plane-being a lasting line is the mode of apprehending 
a plane, and a lasting plane (which is a plane-being's 
solid) is the mode of apprehending our solids. In the 
same way, the Blue line, as it passes through his plane, 
gives rise to a point This point lasts for a minute; 
whereas the Grold point only lasted for a flash. 



CHAPTER III. 

FOUR-SPACE. GENESIS OF A TESSARACT. ITS REPRE* 
SENTATION IN THREE-SPACE. 

Hitherto we have only looked at Model i. Thisi with 
the n^xt seveni represent what we can observe of the 
simplest body in. Higher Space. A few words will 
explain their constructioa A point by its motion traces 
a line. A line by its motion traces either a longer line 
CH* an area ; if it moves at right angles to its own direc* 
tion, it traces a rectangle. For the sake of simplicity. 
we will suppose all movements to be an inch in length 
and at right angles to each other. Thus, a point moving 
iraebs a line an inch long; a line moving traces a square 
inch ; a square moving traces a cubic inch. In these 
cases each of these movements produces something in- 
trinsically different from what we had before. A square 
is not a longer line, nor a cube a lai^ger square. When 
the cube moves, we are unable to see any new direction 
In which it can move, and are compelled to make it 
move in a direction which has previously been used. 
« Let us suppose there is an unknown direction at right 
angles to ^1 our known directions, just as a third 
direction would be unknown to a being confined to the 
surface of the table. And let the cube move in this 
unknown direction for an inch. We call the figure it 
traces a Tessaract The models are representations 
of the appearances a Tessaract would present to us if 
shown in various ways. . Consider for a moment what 
happens to a square when moved to form a cube; Each 
of ito line% moved in the new direction, traces a square ; 



Four-Space. Genesis of a Tessaract. 119 



' the square itself traces a new figure, a cube, which ends 
in another square. Now, our cube, moved in a new 
direction, will trace a tessaract, whereof the cube itself 
is the beginning, and another cube the end These two 
cubes are to the tessaract as the Black square and White 
square are to the cube. A plane-being could not see 
both those squares at once, but he could make models 
of them and let them both rest in his plane at once. So 
also we can make models of the beginning and end of 
the tessaract Model i is the cube^ which is ito begin- 
ing ; Model 2 is the cube which is ito end It will be 
noticed that there are no two colours alike in the two 
models. The Silver point corresponds to the Gold point, 
that is, the Silver point is the termination of the line 
traced by the Gold point moving in the new direction* 
The sides correspond in the following manner :— 

•Sides. 

Model I. Modilt. 

Black corresponds lo Bri^ht-i^reen 
• ' White „ n Light-grejr 

Vermilion „ « Indian-red 

Blue-green ^ it Yellow-ochre 

Dark-blue „ „ Burnt-sienna 

Light-yellow „ ^ l^^n 

The two cubes should be looked at and compared long 
enough to ensure that the corresponding sides can be 
found quickly. Then there are the following correspon- 
dencies in pointo and lines. 

Points. 

Modd I. Modd s. 

Gold c or re sp onds to Silver 

Fawn „ „ • Ttorquoise 

Tem-ootta „ ^ Earthen 

Buff M » Bhietiat 

Light-blue I, I, Quaker-green 

DuU-purple ^ ^ Peacock-Uue 

Deep*blae » » Oiange-Yemilion 

Red ,t '„ Pniple 



I20 



Niw Era of Thought. 
Lines. 



Model I. 






Orangt 


eorretpoadf to 


CninsoB 


H 


ff 


Green-grajr 


If 


ft 


Blue 


n 


ff 


• Brown 


ft 


ff 


Frencb-grejr 


M 


ff 


Dark-tlate 


n 


ff 


Green 


ft 


ff 


Reddish 


ff 


ft 


Bright-Miie 


H 


ft . 


Leaden • 


ff 


ff 


Deep-ydknr 


If 


If 



Mcdilt. 
LeAf«green 
Diill*green 
Dark-purple 
Purple-brown 
Dull-blue 
Dark-pink 
Pale-pink 
Indigo 
Brown-green 
Dark-green 
Pale-yellow 
Daik 

The colour of the cube itself is invisible, as it is 
covered by its boundaries. We suppose it to be Sage- 
peen. 

These two cubes are just as disconnected when looked 
at b)r us as- the black and white squares would be to a 
plane-being if placed side by side on his plane. He 
cannot see the squares in their right position with regard 
to each other, nor can we see the cubes in theirs. 

Let OS now consider the vermilion side of Model i. 
If it move in the X direction, it traces the cube of. 
Model I. Its Gold point travels along the Orange line, 
anditseU; after tracing the cube^ ends in the Blue-green 
square. But if it moves in the new direction, it will 
'also trace a cube, for the new direction is at right angles 
to the up and away directions^ in which the Brown and 
Blue lines run. Let this square, then, move in the 
unknown direction, and trace a cube. This cube we 
cannot se^ because the unknown direction runs out of 
our space at onoe^ just as the up direction runs out of 
the plane of the taUe. But a plane-being could see the 
square^ ndiich the Blue line traces when moved upwards^ 
by tibe cube being turned round the Blue line, the 



Four-Space. Genesis of a TessaracL 121 

Orange line going upwards ; then the Brown line comes 
into the plane of the table in the -X direction. So 
also with our cube. As treated above, it runs from the 
Vermilion square out of our space. But if the tcssaract 
were turned so that the line which runs from the Gold 
point in the unknown direction lay in our space, and the 
Orange line lay in the unknown direction, we could then 
see the cube which is formed by the movement of the 
Vermilion square in the new direction. 

Take Model 5* There is on it a Vermilion square. 
Place this so that it touches the Vermilion square on 
Model I. All the marks of the two squares are 
identical. This Cube 5, is the one traced by the 
Vermilion square moving in the unknown direction. In 
Model 5, the whole figure, the tessaract, produced by 
the movement of the cube in the unknown direction, is 
supposed to be so turned that the Orange line passes 
into the unknown direction, and that the line which 
went in the unknown direction, runs opposite to the old 
direction of the Orange line. Looking at this new cube^ 
we see that there is a Stone line running to the left from 
the Gold point. This line is that which the Gold point 
traces when moving in the unknown direction. 

It is obvious that, if the Tessaract turns so as to show 
us the side, of which Cube 5 is a model, then Cube i will 
no longer be visible. The Orange liiie will run in the 
unknown or fourth direction, and be out of our sight, 
tc^ether with the whole cube which the Vermilion 
square generates, when the Gold point moves along the 
Orange line. Hence^ if we consider these models as real 
portions of the tessaract, we must not have more than 
one before us at once. When we look at one, the others 
must necessarily be beyond our sight and touch. But 
we may consider them simply as models, and, as such, 
we may let them lie alongside of each other. In this 



122 



New Era of Thmght. 



case^ we must mneniber that their real relationships are 
not those in which we see them. 

We now enumerate the sides of the new Cube 5, so 
that; when we refer to it» any colour may be recognised 
byname; 

The square Vermilion traces a Pale-green cube, and 
ends in an Indian*red square. 

(The colour Pale*green of this cube is not seen, as it 
is entirely surrounded by squares and lines of colour.) 

Each Line traces a Square and ends in a Line. 

TbeBloe ]i&e\. /Light-hrowa square^ 2 

Brown M B Yellow »» 

] Light-red „ 
\ Deep-crimson ,, 



^Deep-yeUow^ 
Green Ǥ 



1 

)l 



Purple-brownlioe 
Duli-blue n 
Dark » 

Jndigo n 



Each Point traces a Line and ends in a Point 



point 



Silver 

Blue-tint n 

Quaker-green m 

Purple 9t 



neGold point) /Stone line^ 

M Buff n I Light-green 

^ •Light-Me „ \ Rich-red 

« Red n ) lEmerald 

It will be noticed that besides the Vermilion square of 
this cube another square of it has been seen before. A 
moment's comparison with the experience of a plane- 
being will make this more clear. If a plane-being has 
before him models of the Black and White squares of the 
Cubc^ he sees that all the colours of the one are different 
.from all the colours of the other. Next, if he looks at 
a model of the Vermilion square, he sees that it starts 
from the Bhie line and ends in a line of the White square, 
the Deep-yellow line. In this square he has two lines 
which he had before^ the Blue line with Gold and Buff 
pointy the Deep^yellow line with Light-blue and Red 
points. To him the Black and White squares are his 
Modeb I and 3, and the. Vermilion square is to him as 
our Model 5 is to us. The left-hand square of Model 5 
is Indian-fed^ and is Uentical. with that of the same 



Four-Space. Genesis of a Tessaract. \2% 

colour on the left-hand side of Model 2. In fact, Model 
5 shows us what lies between the Vermilion face of i, 
and the Indian-red face of 2. 

From the Gold point we suppose four perfectly in- 
dependent lines to spring forth, each of them at right 
angles to all the others. In our space there is only 
room for three lines mutually at right angles. It will 
be found, if we try to introduce a fourth at right angles 
to each of three, that we fail; hence, of these four 
lines one must go out of the space we know. The 
colours of these four lines are Brown, Orange, Blue, 
Stone. In Model i are shown the Brown, Orange, and 
Blue. In Model 5 are shown the Brown, Blue, and 
Stone. These lines might have had any directions at 
first, but we chose to begin with the Brown line going 
up, or Z, the Orange going X, the Blue going Y, and the 
Stone line going in the unknown direction, which we 
will call W. 

Consider for a moment the Stone and the Orange 
lines. They can be seen together on Model ^ by look- 
ing at the lower face of it They are at right angles to 
each other, and if the Orange line be turned to take 
the place of the Stone line, the latter will run into the 
negative part of the direction previously occupied by 
the former. This is the reason that the Models 3, 5, 
and 7 are made with the Stone line always running in 
the reverse direction of that line of Model z, which is 
wanting in each respectively. It will now be easy to 
find out Models 3 and 7. All that has to be done is, to 
discover what faces they have in common with i and 2, 
and these faces will show from which planes of i they 
are generated by motion in the unknown direction. 

Take Model 7. On one side of it there is a Dark- 
blue square, which is identical with the Dark-blue 
square of Model !• Placing it so that it coincides with 



"4 



New Era of Thought. 



I by this iquaie line for line, we see that the square 
neaiest to us is Bumt-sienna» the same as the near 
squaie on Model 2. Hence this cube is a model of 
what the Dark-blue square traces oi| moving in the 
unknown directum Here the unknown direction co- 
incides with the negative away direction. In fact, to 
see this cubc^ we have!been obliged to suppose the Blue 
line turned into the unknown. direction, for we cannot 
look at moie than three of these rectangular lines at 
once in our space, and in this Model 7 we have the 
Bfowo, Orange, and Stone lines. The faces, lines, and 
points of Cube 7 can be identified by the following list. 

The Dark-blue square traces a Dark-stone cube 
(^rfM)se interior is rendered invisible by the bounding 
tquaies), and ends in a Burnt-sienna square 

Each Line traces a Square and ends in a Line. 



The Oiange line' 
^ •Brown ' „ 
,, Frendirgrejr n 
„ Reddish „ 



Azare 
Yellow 
Yellow-green 
,Ochre 



square 



j (Leaf-green 
DtiU-blue 



line 



Dark-pink 
Brown-green 



Each Point traces a Line and ends in a Point 



ponif 



The Gold 
ft Fawn „ 

^ Light-bhie „ 
^ DuU-parple ,» 



line 



5 f Silver 
Turquoise 



point 



Quaker-green „ 
Peacock-blue » 



^Stone 
Smoke 
Ricb-red 
^Green-blue 

If we now take Model 3, we see that it has a Black 
squaie uppermost, and has Blue and Orange lines. 
Hence^ it evklently proceeds from the Black square in 
Model I ; and it has in it Blue and Orange lines, which 
piticeed fimn the Gold point But besides Aese, it has 
running downwards a Stone line. The line wanting is 
the Brown linc^ and, as in the other cuses, when one of 
the three lines of Model i turns out into the unknown 
diiectioii» the Stone line turns into the directfon op- 
posite to that from which the line has turned Take 



Four-Space^ Genesis of a Tessaract 125 



this Model 3 and place it underneath Model i, raising 
the latter so that the Black squares on the two coincide 
line for line. Then we see what would come into our 
view if the Brown line were to turn into the unknown 
direction, and the Stone line come into our space down- 
wards. Looking at this cube, we see that the following 
parts of the tessaract have been generated. 

The Black square traces a Brick^red cube (invisible 
because covered by its own sides and edges), and ends 
in a Bright-green square. 

Each Line traces a Square and ends in a Line. 



The Orange 
„ Crimson 
„ Green-grey 
n Bine 



line 



Asure 
Rose 
Sea-blue 
.Light-I 



square^ 



Leaf-green 
Dull-green 
Dark-purple 
Porple^browa 



line 



Each Point traces a Line and ends in a Point. 



The Gold 
t> Fa*ra 
„ Terra-cotta 
M Buff 



point; 



»» / 



IlneU- 



Silver 
Turquoise 
Earthen 
Blue-tint 



point 



Stone 

Smoke „ 

Magenu ^ 

^Lightrgreen „ ^ 

This completes the enumeration of the regions of 
Cube 3. It may seem a little unnatural that it should 
come in downwards ; but it must be remembered that 
the new fourth direction has no more relation to up-and- 
down than to right-and-left or to nearrand-far. 

And if, instead of thinking of d plane-being as living 
on the surface of a tables we suppose his world to be the 
surface of the sheet of paper touching the Dark-blue 
square of Cube i, then we see that a turn round the 
Orange line, which makes the Brown line go into the 
plane-being's unknown direction, brings the Blue line 
into his downwards direction. 

There still remain to be described Models 4, 6, and!& 
It will be shown that Model 4 is to Model 3 what 
Model 2 is to Model i. That is, if, when 3 is in our 



126 



New Era of Thought. 



spacer it be moved so as to trace a tessaract» 4 will be 
the opposite cube in which the tessaract ends. There 
is no colour common to 3 and 4. Similarly, 6 is the 
opposite boundary of the tessaract generated by 5, and 
8 of that by 7. 

A little closer consideration will reveal several points. 
Looking at Cube 5, we see proceeding from the Gold 
point a Brown, a Blue, and a Stone line. The Orange 
]ine is wanting ; therefore, it goes in the unknown 
direction. If we want to discover what exists in the 
unknown direction from Cube 5, we can get help from 
' Cube I. For, since the Orange line lies in the unknown 
direction from Cube 5, the Gold point will, if moved 
along the Orange line, pass in the unknown direction. 
So also^ the Vermilion square, if moved along in the 
direction of the Orange line, will proceed in the un- 
known direction. Looking at Cube i we see that 
the^ermilfon square thus moved ends in a Blue-green 
square. Then, looking at Model 6, on it, corresponding 
to the Vermilion squkre on Cube 5, is a Blue^green 
square. 

Cube 6 thus shows what exists an inch beyond 5 
in the unknown direction. Between the right-hand 
face on 5 and the right-hand face on 6 lies a cube, the 
one which is shown in Model I. Model i is traced by 
• the Vermilion square moving an inch along the direc- 
tion of the 'Orange line. In Model 5, the Orange line 
goes in the unknown direction ; and looking at Model 6 
we see what we should get at die end of a movement of 
one inch in that direction. We have still to enumerate 
the colours of Cubes 4, 6, and 8, and we do so in the 
following list In the first column is designated the 
part of the cube ; in the columns under 4, 6, 8, come the 
colours which 4^ 6^ 8, respectively have in the parts 
designated in the corresponding line in the first column. 



Four-Space. Genesis of a Tessaract. 127 



4 
Chocolate 



6 
Oak-yellow 



8 
Salmon 



Cube itself:- 

Squares : — 

Lower lace Light-grey 

Upper White 

Left-hand Light-red 

Right-hand Deep-brown 

Near Ochre 

Far Deep-green 

Lines : — 

On ground, going round the square from left 
right ;— 

^4 6 8 

1. Brown-green Smoke 

2. Dark-green Crimson 

3. Pale-yellow Magenui 

4. Dark DuU-green 



Rose 

Deep-brown 

Yellow-ochre 

Blue-green 

YeUow-green 

Dark-grey 



Sea-Uue 

Deep-green 

Deep-crimson 

Dark-grey 

Dun 

Light-yellow 



to 



Dark-purple 
Magenta 
Creen-grey 
Light-green 



Vertical, going round the sides from left to right :~ 
I. Rich-red Dark-pink Indigo 

a. Green-blue French-grey Pale-pink 

5. Sea-green Dark-slate . Dark-slate 

4. Emerald Pale-pink Green 

Round upper face in same order : 
I. Reddish Green-blue 

3. Bright-blue Bright-blue 

3. Leaden Sea-green 

4. Deep-yellow Daik-green 



Points : — 

On lower face, going from left to 
I. Quaker-green Turquoise 

3. Peacock-blue Fawn 

3. Orange-vennilion Terra-cotta 

4. Puiple Earthen 

On upper face ; 
I. Light-blue 
1. DuU-purple 
3. Deep-blue 
4- Red 



Pale-yellow 
Sea-green 
Leaden 
Emerald 

r^ht : — 
Blue-tint 
Earthen 
Terra-cotU 
Buff 



Peacock-blue Purple 

Dull-purple Orange-vermilion 

Deep*blue beep-blne 

Onuige*vmnilkMi Red 



138 



New Era of Thought. 



f 



If any one of these cubes be taken at random, it is 
easy enough to find out to what part of the Tessaract 
it belongs. In all of them, except 2, there will be one 
face^ which is a copy of a face on i ; this face is, in 
fact; identical with the face on i which it resembles. 
And the model shows what lies in the unknown 
direction from that face. This unknown direction is 
turned into our spacer so that we can see and touch the 
fesult of moving a square in it And we have sacrificed 
cme of the three origrinal directions in order to do this. 
It will be found* that the line, which in i goes in the 
4th direction, in the other models always runs in a 
nqgative direction. 

Let us take Model 8, for instance. Searching it for 
a face we know, we come to a Light-yellow face away 
from us. We place this face parallel with the Light- 
ydlow face on Cube i, and^we see that it has a Green 
line going up^ and a Green-grey line going to the right 
from the Bufi* point In these respects it is identical 
with the Light*yellow*face on Cube i. But instead of 
a Blue line coming towards us from the Buff point, 
there is a Light^green line. This Light-green line, then, 
is that which proceeds in the unknown direction from 
the Buff point The line is turned towards us in this 
Model 8 in the negative Y direction ; and looking at 
- the model, we see exactly what is formed when in the 
motion of* the whole cube in the unknown direction, 
the Light^yellow face is moved an inch in that direction. 
It traces out a Salmon cube (v. Table on p. 127), and it 
has SeapUue and Deep-green sides below and above^ 
and Deep-crimson and Dark-grey sides left and right, 
and Dun and Light-yellow sides near and fan If we 
want to verify the correctness of any of these details, we 
mint turn to Modek i and x What lies an inch from 
the Light-yellow square id. the unknown direction? 



Four-Space. Genesis of a Tessaract. 

•31 
Model 2 tells us, a Dun square Now, looking at 8 
JJT see that tow^^^^ Th^fawhat 

iwanmchm the unknown direction from the Liffht- 
ydlow square It is here turned to face us, and we 

^ua.r ^^^ ""^ ^^^"^ '' "^ *^ Light-ydlow 



ia8 



CHAPTER IV. 

TESSARACT MOVIKG THROUGH THREE^PACE. 

MODELS OF THE 8ECTI0K& 

IH Older to obtain a clear conception of the higher 
SLrlTc^in amount of familiarity with the^facU 
Smn in these modeU fa necessary. But the best way 
irS«"nine a systematic Icnowlcdge fa shown here- 
ir wSft UieTmodefa enable us to do. fa to take a 
fLl«^cw^ the subject In all of them we see 
K^r^ndarie, of the tessaract in our space; 
iTlnvo more se, or touch the tessaract s»ol.d.ty 
JL a plane-being can touch the cube's sohdity. 
"1^ «main the four modefa 9. .o. . 1. «. Moddl9 
«M^t« what Ues between i and z If i be moved 
2^ in^ unknown direction, it traces out the 
SJ^^«dai»»- But. obviously, between I and 
TS^^ be an infinite number of exactly s.m.Ur 
• Mlid sectipns ; these are all like Model 9. 

TTakeAe ^ of a plane-b«?ng on the Uble. He 
Jtl SLi^-arc-Uut fa. he sees the lin«. rm.nd 

S^ he knSrs that, if it "^^ J^'^'t^Z^Z 
mZlttioiis direction, it traces a new kmd of figure, the 
2SSrb«^Srwh««»f 5« th« White square If, 

2Sr£ hS mSd. of tiie White and Blade square, 
STLlcSi him the end «Ki beginning of our «b^ 
n rLt^M^ these sauaies aie any number of others, 



Tessamut Mwimg in ThrteSpau. 131 

are. The interior of each is a Light-buff (the colour 
of the substance of the cube), the sides are of the colours 
of the vertical faces of the cabe» and the points of the 
colours of the vertical lines of the cube; vis., Dark^blue, 
Blue-green» Light-yellow, Vermilion lines, and Brown, 
French^fprey, Dark-slate, Green points. Thus^ the square, 
in moving in the unknown direction, traces out a 
succession of squares, the assemblage of which makes 
the cube in la)rers. So also the cube, moving in the 
unknown direction, will at any point of its motion, 
still be a cube ; and the assemblage of cubes thus placed 
constitutes the tessaract in layers. We suppose the 
cube to change its colour directly it begins to move. 
Its colour between i and 2 we can easily determine 
by finding what colours its different parts assume, as 
they move in the unknown direction. The Gold point 
immediately begins to trace a Stone*line, We will 
look at Cube 5 to see what the Vermilion face becomes ; 
we know the interior of that cube is Pale-green (tf. Table; 
Pi 122). Hence, as it moves in the unknown direction, 
the Vermilion square forms in its course a series of 
Pale-green squares. The Brown line gives rise to a 
Yellow square ; hence; at e^ftxy point of its course in 
the fourth direction, it is a Yellow line, until, on taking 
its final position, it becomes a Dull-blue line. Looking 
at Cube 5, we see that the Deep-yellow line becomes 
a Light-red line, the Green line a Deep Crimson one; 
the Gold point a Stone one, the Light-blue point a 
Rich-red one, the Red point an Emerald one, and the 
Buff point a Light-green one. Now, take the Model 9. 
Looking at the left side of it, we see exactly that Into 
which the Vermilion square is transformed, as it moves 
in the unknown direction. The left side is an exact 
copy of a section of Cube S, parallel to the Vennilioa 
face. 



132 



A New Era of Thought. 



Bat we have only accounted for one side of our 
Model 9. There are five other sides. Take the near 
side corresponding to the Dark-blue square on Cube i. 
When the Dark-blue square moves, it traces a Dark- 
stone cube^ of which we have a copy in Cube 7. Look- 
ing at 7 (v. Table; p. 124), we see that, as soon as the 
Dark-Uoe square begins to move, it becomes of a Dark- 
stone colour, and has Yellow, Ochre, Yellow-green, and 
Aruresides^ and Stone, Rich-red, Green-blue, Smoke 
lines mnning in the unknown direction from it. Now, 
the side of Model 9^ which faces us, has these colours 
tile squares being seen as lines, and the lines as points. 
HWice Model 9 is a copy of what the cube becomes, 
so far as the Vermilion and Dark-blue sides are con- 
cerned, when, moving in the unknown direction, it 
traces the tessaract * 

We will now look at the lower square of our modeL 
It is a Brick-red square, with Azure, Rose, Sea-blue, 
and L%ht-brown lines, and with Stone, Smoke^ Magenta, 
and Light-green points. This, then, is what the Black 
square should change into, as it moves in the unknown 
direction. Let us look at Model 3. Here the Stone 
line; which is the line in the unknown direction, runs 
downwards. It is turned into the downwards direction, 
so that the cube traced by the Black square may be 
* ' in our space. The colour of this cube is Brick-red ; 
the Orange line has traced an Azure; the Blue line a 
Light-brown, the Crimson line a Rose, and the Green- 
grey line a Sea-Uue square. Hence, the lower square 
of Model 9 shows what the Black square becomes, as 
it traces the tessaract; or, in other words, the section 
of Model 3 between the Black and Bright-green squares 
exactly ommspoods to the lower face of Model 9 

Therefor^ it appears that Model 9 is a model of a 
section of the tessaract, that.it is to the tessaract what 



Tissaract Moving in Three-Spact. 133 
J^sq«|« between the Black and White squares is to 

the^ mT nf "^ ''**" ""^ *« h*ve to see what 
CoiJtSl "*'*^' ^ Light-yellow squares of 

4,0; 8. E«* cube can be used as an index for showing 

PMS«. M it moves m the unknown direction till it 
becom^ Cube 2. Thus, what becomes of the vSite 
T^riJr^*'^''^^ F«>mtheLighlbl«^^2 
of .ts White square runs downwards the Rich^^^n^ 

Hnes Is il £^1°'^; Deep-green, and Ught-red 
RvT'ij . ^ Rich-red, Green-blue, Sea-green and 
Em««ld pomts. The colour of the cube is'cSol^? 

tst^wZll^'''" " '''^'''^ This de«^?on 
Tht~ V^ ^^'^ "PP*** '"••'^« Of Model a 

the LS^hf .,'*"'*'". *r ^^^' *ho«« conesj^nding to 
Ae Light-yellow and Blue-green of Cube i. What the 

I. ^u ^ °" ^^^^ ^ The colour of the Ust-named 
fa Oak-yellow, and a section parallel to its BluJ^ 
«de is s«m>unded by Yellow-^n. Da^brownX^" 
grey and Rose line, and by GiSSSue; Smok^ M*.S£ 
and Sea-green points. This is exactly similaVTSl 

JS'LSrl,*"^!'^^ Lastly, thTS'^^'oS^^tf 
theLight-yeJlowsidecanbeseenooModelJ. The^c- 
t.on of the cube is a Salmon «,u«* bound«I by D^St 

Zlm'^r^T""''^-'^'^^ Sea.blueSn«t^ 
-n^T^t' ^f"*"' ***8«»*^ •^d Light-green point^ 
ab^t Ji- "'^•S «° «'*'»«» to «uw« SyquS 
th^hT kI' ^'^f'^'^ve.mplytotdcciiieadi 
the whole cube; a plan,; and the relation of STXte 



134 ^ N€W Era of Tlumght. 

tessaiact to that plane can be told by looking at the 
model, which, starting with that plane, stretches from it 
in the unknown directioa 

We have not as yet settled the colour of the interior 
of Model 91 It is that part of the tessaract which is 
traced out by the interior of Cube i. The unknown 
diiectkui starts equally and simultaneously from every 
point of every part of Cube i, just as the up direction 
starts equally and simultaneously from every point of a 
square. Let us suppose that the cube^ which is Light- 
bufr» changes to a Wood-coUwir direcUy it begins to trace 
thetessaract Then the internal part of the section be- 
twecn I and 2 will be a Wood-Colour. The sides of the 
ModdQareof thegreatest importance. Theyare the 
cokmrofthe SIX cubes, 3»4i St ^7t«nd«- Thecolours 
of I and a are wanting; via. Light-buff and Sage>green. 
Thus the section between i and 2 can be found by its 
wanting the colours of the Cubes i and a. 

Looking at Modds 10^ ii.and \2 in a similar manner, 
tlie reader will find they represent the sections between 
Cubes 3 and 4, Cubes 5 and 6^ and Cubes 7 and 8 re- 
^ectivdy. 



CHAPTER V. 

REPRESENTATION OF THREE-SPACE BY NAMES, AND 
. IN A PLANE. 

We may now ask ourselves the best vray of passing on 
to a clear comprehension of the &cts of higher space. 
Something can be effected by looking at these models ; 
but it is improbable that more than a slight sense of 
anal(^y will be obtained thus. Indeed, we have been 
trusting hitherto to a method which has something 
vidous about it— we have been trusting to our sense of 
what tnust be. The plan adopted, as the serious effort 
towards the comprehension of this subject, is to learn a 
small portion of higher space. If any reader fed a difii« 
culty in the forgoing chapters, or if the subject is to be 
taught to young minds, it is far better to abandon all 
attempt to see what higher space must be, and to learn 
what it is from the following chapters. 



Naming a Piece of Space. 

The diagram (Fig. 6) represents a block of 27 cubes, 
which form Set i of the 81 cubes. The cubes are 
coloured, and it will be seen that the colours are ar- 
ranged after the pattern of Model i of previous chapters, 
which will serve as a key to the block. In the diagram, G. 
denotes Gold, O. Orangey F. Fawn, Br. Brown, and so on. 
We will give names to the cubes of this block. They 

>3f 



136 



A Niw Era of TlumglU. 



should not be learnt, but kept for reference. We will 
write these names in three sets, the lowest consisting of 
the cubes which touch the table, the next of those im- 
mediately above them, and the third of those at the top. 
Thus the GkAA cube is called Corvus^ the Orange, Cuspis, 
the Fawn, Nugae, and the central one below, Syce. The 
ooiTCsponding colours of the following set can easily be 
traced*' 



OIns 
Via 
Ilex • 


Semiu 

Mel 

Callis 


Lama 

Iter 

Son 


liucina 

Alvnt 

ArclM 


Mnrex 

Mala 

Moma 


Daps 
Praes 
Far 


Cista 

Dm 

Corvof 


Cadtts 

Syce 

Cuspis 


Crus 

Bolus 

Nugx 



Thus the central or Light-buff cube is called Mala ; the 
middle one of the lower face is Syce ; of the upper face 
Mel ; of the right face, Proes ; of the lel^ Alvus ; of the 
front, Mcena (the Dark-blue square of Model i) ; and of 
the back, Murex (the Light»yellow square). 

Now, if Model i be taken, and considered as represent- 
inga block of 64 cubes, the Gold comer as one cube, the 
Orange line as two cubes, the Fawn point as one cubc^ 
the Dark-blue square as four cubes, the Light-buff interior 
as eight cubes, and so on, it will correspond to the dia- 
grain (F^. 7). This block differs from the last in the 
number of cubes, but the arrangement of the colours is 
the same. The following table gives the names which 
we will use for these cubes. There are no new names ; 
they are only applied more than once to all cubes of the 
icoloufo 



^ 



L.BL 



Bt 



om. 



B.R 



r.G« 



/ 



Fi. <. 




Lll. 


R 


A 


•.r. 


Br 


o.n 


aBi. 


r.ft. 


Bt. 


0.B1. 


0.81. 


CCt 


C 








r. 



r.« 7. 



A 


J — /If/' 


<^ y ^ ^ / J 


^ / / / r A 


^ ^ ^ / ' /LH 


/ / / / / An 


UBl 


ft 


ft 


ft 


••'./; 


Bt. 


OBI 


IB 


BlSI 


r.^/. 


St. 


OLBI 


Ul 


m. 


fo.// 


It. 


Oil 


a»i 


OJl 


rc/f 











'J 



r««- 



in>«/. ijt. 



Three-Spaa by Nanus, and in a Plane. 137 





Olus 


Semiu 


Semita 


Lama 


Fourth 


Via 


Mel 


Mel 


. Iter 


Floor. 


Via 


Mel 


Mel 


Iter 




Ilex » 


CaUis 


Callis 


Sort 




r Bucina 


Murex 


Murex 


Daps 


Third 


)a1vus 


Mala 


Mala 


Proes 


Floor. 


)Alvtts 


Mala 


Mala 


Proes 




^Arctos 


Moena 


Moena 


Far 




( Bucina 


Murex 


Murex 


Daps 


Second 


JAlvus 


Mala 


Mala 


Proes 


Floor, 


)Alvuf 


Mala 


MaU 


Proes 




lArctos 


Moena 


Moena 


Far 


j 


rCisU 


Cadus 


Cadttt 


Crus 


Fiwt ; 


|Dos 


Syce 


Syce 


* Bolus 


Floor, ' 


|Doi 


Syce 


Syce 


Bolus 


( 


. Corvus 


Cttspif 


Cuspis 


Nugae 



If we noW consider Model i to represent a block, five 
cubes each way, built up of inch cubes, and colour it in 
the same way, that is, with similar colours for the corner* 
cubes, edge-cubes, face-cubes, and interior-cubes, we 
obtain what is represented in the diagram (Fig. 8). 
Here we have nine Dark-blue cubes called Moena ; that 
is^ Mcena denotes the nine Dark-blue cubes, forming a 
layer on the front of the cube^ and filling up the whole 
front except the edges and points. Cuspis denotes three 
Orange, Dos three Blue, and Arctos three Brown cubes. 

Now, the block of cubes can be similarly increased to 
any size we please. The comers will always consist of 
single cubes ; that is, Corvus will remain a single cubic 
inch, even though the block be a hundred inches each 
way. Cuspis, in that case, will be 98 inches long, and 
consist of a row of 98 cubes ; Arctos^ also, will be a long 
thin line of cubes standing up. Moena will be a thin 
layer of cubes almost covering the whole front of the 
block ; the number of them will be 98 times 98. Sytit' 



138 



A New Era of Thought. 



will be a similar square layer of cubes on the ground^ so 
also Mely Alvus, Proes, and Murex in their respective 
places. Mala, the interior of the cube» will consist of 
98 times 98 times 98 inch cubes. 

Now, if we continued in this manner till we had a 
very laige block of thousands of cubes in each side 
Corvus would, in comparison to the whole block, be a 
minute point of a cubic shape, and Cuspis would be a 
mere line of minute cubes, which would have length, but 
very small depth or height Next, if we suppose this 
much sub-divided block to be reduced in size till it be- 
comes one measuring an inch each way, the cubes of 
which it consists must each of them beoome extremely 
minute^ and the comer cubes and line cubes would be 
scarcely discernible. But the cubes on the faces would 
be just as visible as before. For instance, the cubes com- 
posing Moena would stretch out on the face of the cube 
to as to fill it up. They would form a layer of extreme 
thinness, but would cover the face of the cube (all of it 
except the minute lines and points). Thus we may use 
the words Corvus and Nugse, etc, to denote the comer- 
points of the cube, the words Moena, Syce, Mel, AlvuSp 
Proes, Murex, to denote the faces. It must be remem- 
bered that these faces have a thickness, but it is ex- 
tremely minute compared with the cube. Mala would 
denote all the cubes of the interior except those, which 
compose, the faces, edges, and points. Thus, Mala would 
practically mean tiie whole cube except the colouring on 
it And it is in this sense that these words will be used 
In the models, the Gold point is intended to be a G)rvus, 
only it is made large to be visible ; so too the Orange 
line is meant for Cuspis, but magnified for the same 
reasoa Finally, the vj names of cubes, with which we 
b^an, oome to be the names of the points, lines, and 
faces of a cube^ as shown in the diagram (Fig. 9). With 




iT0j(mp* tjl^ 



Three-Space 6y Names, and in a Pla$u. 139 



these names it is easy to express what a plane-being 
would see of any cubCi Let us suppose that Mcena is 
only of the thickness of his matter. We suppose his 
matter to be composed of particles, which slip about on 
his plane^ and are so thin that he cannot by any means 
discern any thickness in them* So he has no idea of 
thickness. But we know that his matter must have some 
thicknessi and we suppose Mcena to be of that degree of 
thickness. If thecube be placed so that Mcena is in his 
plancp Corvusi Cuspis, Nugae^ Far^ Sors, CalliSi Ilex and 
Arctos will just come into his apprehension ; they will be 
like bits of his matter, while all that is beyond them in 
the direction he does not know, will be hidden from hinu 
Thus a plane-being can only perceive the Mcena or Syce 
or some one other face of a cube ; that is, he would take 
the Moena of a cube to be a solid in his plane-space, and 
he would see the lines Cuspis, Far, Callis, Arctos. To him 
they would bound it The points Corvus, Nugae,'Sors, 
and Ilex, he would not see, for th^ are only as long as 
the thickness of his matter, and that is so slight as to be 
indiscernible to him. 

We must now go with great care through the exact 
processes by which a plane-being would study a cubCi 
For this purpose we use square slabs which have a cer- 
tain thickness, but are supposed to be as thin as a plane- 
being's matter. Now, let us take the first set of 81 cubes 
again, and build them from i to 27. We must realize 
clearly that two kinds of blocks can be built It may 
be built of 27 cubes, each similar to Model i, in which 
case each cube has its rq^ions coloured, but all the cubes 
are alike. Or it may be built of 27 differently coloured 
cubes like Set i, in which case each cube is coloured 
wholly with one colour in all its regions. If the latter 
set be used, we can still use the names Mcena, Alvus, etc 
to denote the front, side, etc, of any one of the cubes, 



I40 



A New Era of Thought. 



whatever be its colour. When they are built up, place 
a piece of card against the front to represent the plane 
cm which the plane-being lives. The front of each of 
the cubes in the front of the block touches the plane. 
In previous chapters we have supposed Mcena to be a 
Blue square. But we can apply the name to the front 
of a cube of any colour. Let us say the Moena of each 
front cube is in the plane ; the Mcena of the Gold cube 
is Gold, and so on. To represent this, take nine slabs 
of the same colours as the cubes. Place a stiff piece of 
cardboard (or a book-cover) slanting from you, and put 
the slabs on it They can be supported on the incline 
so as to prevent their slipping down away from you by 
a thin book, or another sheet of cardboard, which stands 
for the surface of the plane-being's earth. 

We will now give names to the cubes of Block i of 
the 8 1 Set We call each one Mala, to denote that it is 
u cube. 'They are written in the following list in floors - 
or layers, and are supposed to run backwards or away 
from the reader. Thus, in the first layer, Frenum Mala 
is behind or farther away than Urna Mala; in the 
second la}rer, Ostrum is in front, Uncus behind it, and 
Ala behind Uncus. 



Third, or i Man Mala 
Top < Spicula Mala 
FkM>r. (ComesMala 

Secoiid,or ( Ala Mala 
Middle < Uncos Mala 
Floor. I Ostnim Mala 

Ftfst, or / Sector Mala 
Bottom < Frenum Mala 
Floor. lUraaMala 



Merces Mala 
Mora Mala 
Tibicen Mala 

Conis Mala 
PaUor Mala 
Bidens Mala 

HamaMala 
PlebtMaU 
Moles MaU 



Tyro Mala 
Oliva Mala 
Vestis Mala 

Aer Mala 
Tei^m Mala 
ScenaMala 

Remus Mala 
Sypho Mala 
Saltus Mala 



These names should be learnt so that tlie dtflferent 
cubes in die block can be referred to quite OMily and 



■\ 



I 



-? 



-X' 



FglO. 



Utm 



OblfM 



[r#>«/.i4i* 



Thne-Spau by Natnes^ and in a Plam. 141 

immediately by name. They must be learnt in every 
order, that is, in each of the three directions backwards 
and forwards, e.g, Urna to Saltus, Urna to Sector, Uma 
to Comes ; and the same reversed, viz.. Comes to Urna, 
Sector to Urna, etc Only by so learning them can 
the mind identify any one individually without even a 
momentary reference to the others around it It is well 
to make it a rule not to proceed from one cube to a 
distant one without naming the intermediate cubes. 
For, in Space we cannot pass from one part to another 
without going through the intermediate portions. And, 
in thinking of Space, it is well to accustom our minds to 
the same limitations. 

Uma Mala is supposed to be solid Gold an inch each 
way ; so too all the cubes are supposed to be entirely of 
the colour which they show on their faces. Thus any 
section of Moles Mala will be Orange^ of Plebs Mala 
Black, and so on. 

Let us now draw a pair of lines on a piece of paper 
or cardboard like those in the diagram (Fig. 10). In 
this diagram the top of the page is supposed to rest on 
the table, and the bottom of the page to be raised and 
brought near the eye^ so that the plane of the diagram 
slopes upwards to Uie reader. Let Z denote the upward 
direction, and X the direction from left to right Let 
us turn the Block of cubes with its front upon this 
slope ue. so that Uma fits upon the square marked 
Umal Moles will be to the right and Ostnim above 
Uma, ie. nearer the eye. We might leave the block as it 
stands and put the piece of cardboard against it ; in this 
case our plane-world would be vertical It is difficult to 
fix the cubes in this position on the plane^ and therefore 
more convenient if the cardboard be so inclined that 
they will not slip off. But the upward direction must 
be identified with Z. Now, ^taking the siab% let us 



142 



A New Era of Tlumght. 



compose what a plane-being would see of the Block. 
He would perceive just the front faces of the cubes of 
the Block, as it comes into his plane ; these front faces 
we may call the Moenas of the cubes. Let each of the 
slabs-represent the Moena of its corresponding cube, the 
GoM slab of the Gold cube and so on. They are thicker 
than they should be ; but we must overlook this and 
suppose we simply see the thickness as a line. We thus 
buikl a square of nine slabs to represent the appearance 
to a plane-being di the front face of the Block. The 
mkldle one» Bidens Moena, would be completely hidden 
from him b^ the others on all its sides, and he would 
see the edges of the eight outer squares. If the Block 
now begin to move through the plane, that is, to cut 
thiDiigfa the piece of paper at right angles to it, it will 
not for some time appear any different For the sections 
of Uma are all Gold like the front face Moena, so that 
tiie appearance of Uma at any point in its passage will 
be a Gold square exactly like Uma Moena, seen by the 
plane-being as a line Thus, if the speed of the Block's 
passage be one inch a minute^ the plane-being will see 
no change for a minute. In other words, this set of 
slabs lasting one minute will represent what he ^t!^ 
. When the Block bai passed one Ineht a diflercnt set 
of cubes appears. Remove the front layer of cubes. 
There will now be in contact with the paper nine new 
cnbes^ whiose names we write in the order in which we 
sliould see them through a piece of glass standing up- 
r^ in front of the Block : 



^icnlaMala 


MonMaU 


OUvaMaU 


VacuMala 


Pallor Mate 


TeixnmMala 


FfOHimlUk 


PlebaMala 


SyphoMala 



We pick out nine slabs to represent the Moenas Of 
these cttbes^ and placed in order they show what the 



Thret^fMue by Nantes^ and in a Plane. 143 

plane-being sees of the second set of cubes as they pass 
through. Similarly the third wall of the Block will 
come into the plane, and looking at them similarly, as 
it were through an upright piece of glass, we write their 
names: 



Mats Mala 


MercesMala 


Tyro Mala 


Ala Mala 


Cortts MaU 


AerMaU 


Sector Mala 


HamaMaU 


Remus Mala 



Now, it is evident that these slak>s stand at different 
times for different parts of the cubes. We can imagine 
them to stand for the Moena of each cube as it passes 
through. In that case, the first set of slabs, which we 
put up, represents the Moenas of the front wall of cubes ; 
the next set, the Moenas of the second wall. Thus, if 
all the three sets of slabs be tc^^her on the table, we 
have a representation of the sections of the cube. For 
some purposes it would be better to have four sets of 
slabs, the fourth set representing the Murex of the 
third wall ; for the three sets only show the front faces 
of the cubes, and therefore would not indicate anything 
about the back faces of the Block. For instance, if a 
line passed through the Block diagonally from the 
point Corvus (Gold) to the point Lama (Deep-blue), it 
would be represented on the slabs by a point at the 
bottom left-hand comer of the Gokl slab, a second point 
at the same comer of the Light-buff slab, and a third 
at the same comer of the Deep-blue slab. Thus, we 
should have the points mapped at which the line entered 
the fronts of the walls of cubes, but not the point in 
Lama at which it would leave the Block, 

Let the Diagrams i, 2, 3 (Fig. 11), be the three sets 
of slabs. To see the diagrams properly^ the reader must 
set the top of the page on the tables and look along thie 
page from the bottom of it The line in question, which 



J 



144 



A New Era of Thought. 



runs ffom the bottom left-hand near corner to the top 
right-hand far comer of the Block will be represented in 
the three sets of slabs by the points A, B, C To com- 
plete the diagram of its course, we need a fourth set of 
slabs for the Murex of the third wall ; the same object 
might be attainedf if we had another Block of 27 cubes 
behind the first Block and represented its front or 
Moenas by a set of slabs. For t&e point, at which the 
line leaves the ifirst Block is identical with that at which 
it enters the second Block. 

If we suppose a sheet of glass to be the plane-world, 
the Diagrams i, 2, 3 (Fig. 11), may be drawn more 

, naturally to us as Diagrams a, /3, 7 (Fig. xi). Here a 
represents the Moenas of the first wall, /3 those of the 
second, 7 those of the third But to get the plane- 
being's view wc must look over the edge of the glass 
down the Z axis. 

» Set ^ of slabs represent the Moenas of Wall 2. These 
Moenas are in contact with the Murex of Wall i. Thus 
Set 2 will show where the line issues from Wall i as 
well as where it enters Wall 2. 

The plane-being, therefore, could get an idea of the 
Block of cubes by learning these slabs. He ought not 
to call the Gold slab Urna Mala, but Uma Moena, and 
so on, because all that he learns are Moenas, merely the. 
thin faces of the cubes. By introducing the course of 
time^ he can represent the Block more nearly. For, if 

' he supposes it to be passing an inch a minute^ he may 
give the name Uma Mala to the Gold slab enduring for 
a minute. 
But, when he has learnt the slabs in this position and 

sequence^ he has only a vety partial view of the Block. 
Let the Block turn round the Z axis» as Model I turns 

round the Brown line. A different set of cubes comes 

into his plane^ and now they come in on the Alvus 









r^% r— 1 


^^ 




-►X 


^mm 














ft 
















) — ■ 








^ 


z 


0) 




z 


w 




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w 





•»x 



F.^ II. 



A 



C (a) f 



t- 






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(y) 



{nAmp.%u> 



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«l 


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Sit 






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_^ 


















_ 








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ir#^Mr A t4S- 



ThreeSpace by Names, and in a Plane. 145 

faces. (Alvus is here used to denote the left-hand faces 
of the cubes, and is not supposed to be VermiUon ; it is 
simply the thinnest slice on the left hand of the cube 
and of the same colour as the cube.) To represent this» 
the plane-being should employ a fresh set of slabs, for 
there is nothing common to the Moena and Alvus faces 
except an edge. But, since each cube is of the same 
colour throughout, the same slab may be used for its 
different faces. Thus the Alvus of Uma Mala can be 
represented by a Gold slab. Only it must never be 
forgotten that it is meant to be a new slab, and is not 
identical with the same slab used for Moena. 

Now, when the Block of cubes has turned round the 
Brown line into the plane, it is clear that they will be 
on the side of the Z axis opposite to that on which 
were the Moena slabs. The line, which ran Y, now runs 
—X. Thus the slabs will occupy the second quadrant 
marked by the axes, as shown in the diagram (Fig. 13). 
Each of these slabs we will name Alvus. In this view, 
as before, the book is supposed to be tilted up towards 
the reader, so that the Z axis runs from O to his eye. 
Then, if the Block be passed at right angles through the 
plane^ there will come into view the two sets of slabs 
represented in the Diagrams (Fig. 13). In copying this 
arrangement with the slabs, the cardboard on which 
they are arranged must slant upwards to the eye, £/., 
OZ must run up to the e3fe, and the sides of the slabs 
seen are in Diagram 2 (Fig. 13), the upper edges of 
Tibicen, Mora, Merces ; in Diagram 3, the upper edges 
of Vestis, Oliva, Tyra 

There is another view of the Block possible to a plane- 
being. If the Block be turned round the X axis» the 
lower face comes into the vertical plane This corre- 
sponds to turning Model I round the Orange line On 
referring to the diagram (Fig. ^4), we now see that the 



146 



A New Era of Thought. 



name di the faces of the cubes coining into the plane is 
Syce. Here the plane-being looks from the extremity 
of the Z axis and the squares^ which he sees run from 
him in the -Z direction. (As this turn of the Block 
brings- its Syce into the vertical plane so that it ex- 
tends three inches below the base line of its Moena, it 
is evident that the turn is only possible if the Moena be 
originally at a height of at least three inches above the 
plane-being's earth line in the vertical plane.) Next^ if 
the Block be passed through the plane, the sections 
shown in the Diagrams 2 and 3 (Fig. 14) are brought 
into view. 

Thus» there are three distinct ways of r^arding the 
cubic Block, each of them equally primary ; and if the 
plane-being is to have a correct idea of the Block, he 
must be equally familiar with each view. By means of 
the slabs each aspect can be represented ; but we must 
remember in each of the three cases, that the slabs 
represent diflferent parts of the cube. 

When- we look at the cube Pallor Mala in space, we 
see that it touches six other cubes by its six faces. But 
the plane-being could only arrive at this fact by com- 
paring different views. Taking the three Moena sec- 
tions of the Block, he can see that Pallor Mala Moena 
touches Plebs Moena, Mora Moena, Uncus Moena, and 
Tergum^ Moena by lines. And it takes the place of 
Bidens Moena, and is itself displaced by Cortis Moena 
as the Block passes through the plane. Next, this 
same Pallor Mala can appear to him as an Alvus. In 
this case, it touches Plebs Alvus, Mora Alvus, Bidens 
Alvu% and Cortis Alvus by lines, takes the place of 
Uncus Alvus, and is itself displaced by Teiffum Alvus 
as the Block moves. Similarly he can observe the 
relation^ if the Syce of the Block be in his plane 

Hence^ this unknown body Pallor Mala appears to 



Ufiu 



Mmm 



^ -X- 



M«l«« 



Hnm 



H#<fu 



Ufftt 



f« 



r^.i€ 



iT0/m/» 147* 



• ♦ 



Three-Space by Names, and in a Plane. 147 

him now as one plane-figure now as another, and comes 
before him in different connections. Pallor Mala is that 
which satisfies all these relations. Each of them he can 
in turn present to sense ; but the total conception of 
Pallor Mala itself can only, if at all, grow up in his mind. 
The way for him to form this mental conception, is to 
go through all the practical possibilities which Pallor 
Mala would afford him by its various movements and 
tum& In our world these various relations are found 
by the most simple observations ; but a plane-being 
could only acquire them by considerable labour. And 
if he determined to obtain a knowledge of the physical 
existence of a higher world than his own, be must pass 
through such discipline. 



We will see what change could be introduced into the 
' shapes he builds by the movements, which he does not 
know in his world. Let us build up this shape with the 
cubes of the Block : Urna Mala, Moles Mala, Bidens 
Mala, Tibicen Mala. To the plane-being this shape 
would be the slabs, Urna Moena, Moles Moena, Bidens 
Moena, Tibicen Moena (Fig. 15). Now let the Block 
be turned round the Z axis, so that it goes past the 
position, in which the Alvus sides enter the vertical 
plane. Let it move until, passing through the plane, 
the same Moena sides come in again. The mass of the 
Block will now have cut through the plane and be on the 
near side of it towards us ; but the Moena faces only will 
be on the plane-being's side of it The diagram (Fig. 16) 
shows what he will see, and it will seem to him similar 
to the first shape (Fig. 15) in e^txy respect except 
its disposition with regard to the Z axis. It lies in the 
direction —X, opposite to that of the first figure. How- 
ever much he turn these two figures about in the plane, 



148 



A New Era of Thought. 



he cannot make one occupy the place of the other, part 
for part Hence it appears that, if wc turn the plane* 
being's figure about a line, it undergoes an operation 
which is to him quite mysterious. He cannot by any 
turn in his plane produce the change in the figure pro- 
duced by us. A little observation will show that a 
plane-being can only turn round a point Turning 
round a line is a process unknown to him. Therefore 
one of the elements in a plane-being's knowledge of a 
space higher than his own» will be the conception of a 
kind of turning which will change his solid bodies into 
their own images.' 



.♦: 



CHAPTER VI. 

THE MEANS BY WHICH A PLANE-BEING WOULD 
ACQUIRE A CONCEPTION OF OUR FIGURES. 

Take die Block of twenty-seven Mala cubes, and build 
up the following shape (Fig. 18) :— 

Uma Mala, Moles Mala, Plebs Mala, Pallor Mala, 
Mora Mala. 

If this shape^ passed through the vertical plane; the 
plane-being would perceive :— 

(i) The squares Urna Moena and Moles Moena. 

(2) The three squares Plebs Moena, Pallor Moena, 
Mora Moena, 

and then the whole figure would have passed through 
his plane. 

If the whole Block were tumefd round the Z axis till 
the Alvus sides entered, and the figure built up as it 
would be in that disposition of the cubes; the plane-being 
would perceive during its passage through the plane :— 

(i) Uma Alvus; 

(2) Moles Alvus, Plebs Alvus, Pallor Alvus, Mora 
Alvus, which would all enter on the left side of the Z 
axis. 

Again, if the Block were turned round the X axis, the 
Syce side would enter, and the cubes appear in the 
following order :— 

(i) Uma Syce, Moles Syce, Plebs Syce ; 

(2) Pallor Syce ; ^ 

(3) Mora Syce. 

«49 



ISO 



A New Era of Thought. 



A comparison of these three sets of appearances would 
give the plane-being a full account of the figure. It is 
that whidi can produce these various appearances. 

Let us now suppose a glass plate placed in front of 
the Block in its first position. On this plate let the axes 
X and Z be drawn. They divide the surface into four 
parts^ to which we give the following names (Fig. 17) :— 

Z Xathat quarter defined by the positive Z and posi- 
tive X axis. 

ZXsthat quarter defined by the positive Z and 
native X axis (which is called '' Z negative X 'O- 

ZX»that quarter defined by the negative Z and 
9^tive X axi& 

ZXathat quarter defined by the negative Z and 
positive X axis. 

The Block appears in these diflferent quarters or quad- 
rants» as it is turned round the diflferent axes. In Z X 
the Moenas appear, in Z X the Alvus faces» in 2 X the 
Syces. In each quadrant are drawn nine squares, to 
receive the faces of the cubes when they enter. For 
fawtancr» in Z X we have the Moenas of: — 

Z 

Comes Tibtcen Vestis 
Ostrum Btdens Scena 
Uroa Moles Saltus 
X 



And in Z X we have the Alvus of :-^ 



Al 



Vttf 



M 



^fflAf 



^% 






Sy 



€t€ 



(r#/Mr A 190. 



-X- 



Man 

Ala 

Sector 



Sptcttia 

Unctts 

rreniim 



Comes 
Ottrum 
Uma 



And in the^X we have the Syces of :-« 



-Z 



Uma 

riesom 

Sector 



Moles 
Plebs 
Hama 



Saltus 
Sypho 



Fr 



z 




y 






'' 




■7^ 






/ 




, 




/ . 


A 






/ 




/ 


cll— 







/'z* 



%•»« /J) 



z 
z z 




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X 



■ 1 

r ' 



rn 



(t) 



%« 



(i) 



JL. 



|IV/Mf /. tfl. 



Plane^Bnng's Ccnc^tim of <mr Figures. 151 

Now, irthe shape taken at the beginning of this chapter 
' be looked at through the glass, and the distance of the 
second and third walls of the shape behind the glass 
be considered of no account— that is, if they be treated 
as close up to the glass— ^we get a plane outline, which 
occupies the squares Uma Moena, Moles Moena, Bidens 
Moena, Tibicen Moena. This outline is called a pro- 
jection of the figure. To see it like a plane-being, we 
should have to look down on it along the Z axis. 

It is obvious that one projection does not give the 
shape. For instance, the square Bidens Moena might 
be filled by either Pallor or Cortis. All that a square in 
the room of Bidens Moena denotes, is that there is a 
cube somewhere in the Y, or unknown, direction from 
Bidens Moena. This view, just taken, we should call 
the front view in our space ; we are then looking at it 
along the negative Y axis. 

When the same shape is tunied round on the Z axis, 
so as to be projected on the Z X quadrant, we have the 
squares— Uma Alvus, Frenum Alvus, Uncus Alvus, 
Spicula Alvus. \^en it is turned round the X axis, 
and projected on Z X, we have the squares, Uma Syce, 
Moles Syce, Plebs Syce, and no more. TlUs is what is 
ordinarily called the ground plan ; but we have set it in a 
different position from that in which it is usually drawn. 

Now, die best method for a plane-being of* familiar- 
izing himself with shapes in our space, would be to 
practise the realization of them from their different pro- 
jections in his plane. Thus, given the three projections 
just mentioned, he should be able to constract the figure 
from which they are derived. The projections (Fig. 19) 
are drawn below the perspective pictures of the shape 
(Fig. 1 8). From the front, or Moena view, he would 
conclude that the shape was Uma Mala, Moles Mala, 
Bidens Mala, Tibicen Mala ; or instead of these, or also 



152 



A New Era of Thought. 



in addition to them, any of the cubes running in the Y 
direction from the plane* That is, from the Moena pro- 
jection he might infer the presence of. ail the following 
cubes (the word Mala is omitted for brevity): Uma, 
Frenum, Sector, Moles, Plebs^ Hama, Bidens. Pallor, 
Cortis^ Tibicen, Mora, Merces. 

Next, the Alvus view or projection might be given by 
the cubes (the word Mala being again omitted) : Urna, 
Moles, Saltus, Frenum, Flebs, Sypho, Uncu% Pallor, 
Tergum, Spicula, Mora, Oliva. Lastly, looking at the 
ground plan or Syce view, he would infer the possible 
presence of Urna,' Ostrum, Comes, Moles, Bidens^ 
Tibicen, Plebs, Pallor, Mora. 

Now, the shape in higher space, which is usually there, 
is that which is common to all these three appearances. 
It can be determined, therefore, by rejecting those cubes 
which are not present in all three lists of cubes possible 
from the projections. And by this process the plane- 
being could arrive at the enumeration of the cubes 
iHiich belong to the shape of which he had the pn>« 
jections. After a time, when he had experience of the 
cubes (which, though invisible to him as wholes, he 
could see part by part in turn entering his space), the 
projections would have more meaning to him, and he 
might comprehend the shape they expressed fragmen- 
tarily in his space. To practise the realization from 
projections, we should proceed in this way. First, we 
should think of the possibilities involved in the Moena 
view, and build them up ih cubes before us. Secondly, 
we should build up the cubes possible from the Alvus 
view. Again, taking the shape at the b^inning of the 
chapter, we should find that the shape of the Alvus 
possibilities intersected that of the Moena possibilities in 
Urna, Molea^ Frenum, Plebs, Pallor, Mora ; or, in other 
words, these pibes are common to both. Thirdly, we 



:' 



Plane-Beings Conception of our Figures. 153 

should build up the Syce possibilities, and, comparing 
their shape with those of the Moena and Alvus pro- 
jections, we should find Urna, Moles, Plebs» Pallor, Mora, 
of the Syce view coinciding with the same cubes of the 
other viewsi the only cube present in the intersection of 
the Moena and Alvus possibilities, and not present in 
the Syce view, being Frenum. 

The determination of the figure denoted by the three 
projections, may be more easily eflected by treating each 
projection as an indication of what cubes are to be cut 
away. Taking the same shape as before, we have in the 
Moena projection Urna, Moles, Bidens, Tibicen; and 
the possibilities from them are Urna, Frenum, Sector, 
Moles, Plebs, Hama, Bidens, Pallor, Cortis, Tibicen, 
Mora, Merces. This may aptly be called the Moena 
solution. Now, from the Syce projection, we learn at 
once that those cubes, which in .it would produce Frenum, 
Sector, Hama, Remus, Sypho, Saltus, are not in the 
shape. This absence of Frenum and Sector in the Syce 
view proves that their presence in the Moena solution is 
superfluous. The absence of Hama removes the possi- 
bility of Hama, Cortis, Merces. The absence of Remus, 
Sypho, Saltus, makes no difference, as neither they nor 
any of their Syce possibilities are present in the Moena 
solution. Hence, the result of comparison of the Moena 
and Syce projections and possibilities is the shape : 
Urna, Moles, Plebs, Bidens, Pallor, Tibicen, Mora. This 
may be aptly called the Moena-Syce solution. Now, 
in the Alvus projection we see that Ostrum, Comes, 
Sector, Ala, and Mars are absent The absence of 
Sector, Ala, and Mars has no effect on our Moena-Syce 
solution ; as it does not contain any of their Alvus possi- 
bilities. But the absence of Ostrum and Comes proves 
that in the Moena-Syce solution Bidens and Tibicen are 
superfluous, since their presence in the original shape 



154 



A New Era of Thought. 



would give Ostruin and Comes in the Alvus projection. 
Thus we mrive at the Moena-AIvus-Syce solution, 
which gives us the shape: Uma, Moles, Plebs, Pallor, 
Mora. 

It will be obvious on trial that a shape can be instantly 
recognised from its three projections, if the Block be 
thoroughly well known in all three positions. Any 
difiiculty in the realization of the shapes comes from the 
arUtraiy habit of associating the cubes with some one 
direction in which they happen to go with regard to us. 
If we lemembq' Ostrum as above Uma, we are not 
remembering the Block, but only one particular relation 
of the Block to us. That position of Ostrum is a fact 
as much related to ourselves as to the Block. There is, 
of course, some information about the Block implied in 
that position ; but It is so mixed with information about 
pufselves as to be ineffectual knowledge of the Block. 
It is of the highest importance to enter minutely into 
all the details of solution written above. For, corre- 
sponding to eveiy operation necessary to a plane-being 
for the comprehension of our world, there is an opera- 
tion, with which we have to become familiar, if in our 
turn we would enter into some comprehension of a 
world higher than our own. Every cube of the Block 
ought to be thoroughly known in all its relations. And 
the Block must be rq^arded, not as a formless mass out 
oC which ^shapes can be made, but as the sum of all 
possible shapes, from which any one we may choose is a 
selection. In fact, to be familiar with the Block, we 
ought to know eveiy shape that could be made by any 
•election of Its cubes ; or, in other words, we ought to 
make an exhaustive study of it In the Appendix is 
given a set of exereises in the use of these names (which 
Ibnn a language of shape), and In various kinds of pro- 
'The projections studied In this chapter are 



Plane-Beiftg's Conaption of our Figures. 155 

not the only, nor the most natural, prejections by which 
a plane-being would study higher space. But they 
suffice as an illustration of our present purpose. If the 
reader will go through the exercises in the Appendix, 
and form others for himself, he will find every bit of 
manipulation done will be of service to him in the com- 
prehension of higher space. 

There is one point of view in the study of the Block 
by means of slabs, which is of some interest The cubes 
of the Block, and therefore also the representative slabs 
of their faces, can be regarded as forming rows and 
columna There are three sets of them. If we take 
the Moena view, they represent the views of the three 
walls of the Block, as they pass through the plane. To 
form the Alvus view, we only have to rearrange the 
slabs, and form new sets. The first new set is formed 
by taking the first, or left-hand, column of each of the 
Moena sets. The second Alvus set is formed by taking 
the second or middle columns of the three Moena sets. 
The third will consist of the remaining or right-hand 
columns of the Moenas. 

Similarly, the three Syce sets may be formed from 
the three horizontal rows or floors of the Moena sets. 

Hence, it appears that the plane-being would study 
our space by taking all the possible combinations of the 
corresponding rows and columna He would break up 
the first three sets into other sets, and the study of the 
Block would practically become to him the study of 
these various arrangements. 



i 



CHAPTER VII. 

FOUR-SPACE : ITS REPRESENTATION IN THREE- 
SPACE. 

We now come to the essential difficalty of our task 
All that has gone before is preliminaiy. We have now 
to frame the method by which we shall introduce 
through our space-figurl» the figures of a higher space. 
When a plane-being studies our shapes of cubes, he has 
to use squares. He is limited at the outset A cube 
appears to him as a square. On Model i we see the 
various siquares as which the cube can appear to him. 
We suppose the plane-being to look from the extremity 
of the Z axis down k vertical plane. First, there is the 
Moena jKiuare. Then there is the square given by a 
section parallel to Moena, whidi he recognises by the 
variation of the bounding lines as soon as the cube 
bq^s to pass through his plane. Then comes the 
Murex square. Next, if the cube be turned round the 
Z axis and passed through, he sees the Alvus and Proes 
squares and the intermediate section. So too with the 
Syce and Md squares and the section between them. 

Now, dealing with figures in higher space, we are in 
an analogous position. We cannot grasp the element 
of which they are composed. We can conceive a cube ; 
but that which corresponds to a cube in higher space is 
b^ond our grasp. But the plane-being was obliged to 
use two-dimensional figures^ squares, in arriving at a 
notioQ of' a three-dimensional figure ; so also must we 






i 



i 



i 



^•^1 



,: 



Representation of Four-Space. 157 

use three-dimensional figures to arrive at the notion of 
a four-dimensional. Let us call the figure which corre- 
sponds to a square in a plane and a cube in our space, a 
tessaract Model i is a cube. Let us assume a tessa- 
ract generated from it Let us call the tessaract Uma. 
The generating cube may then be aptly called Uma Mala. 
We may use cubes to represent parts of four-space, but 
we must always remember that they are to us, in our 
study, only what squares are to a plane-being with re- 
spect to a cube. 

Let us again examine the- mode in which a plane- 
being represents a Block of cubes with slabs. Take 
Block I of the 81 Set of cubes. The planc-being repre- 
sents this by nine slabs, which represent the Moena face 
bf the block. Then, omitting the soh'dity of these first 
nine cubes, he takes another set of nine slabs to repre- 
sent the next wall of cubes. Lastly, he represents the 
third wall by a third set, omitting the solidity of both 
second and third walls. In this manner, he evidently 
represents the extension of the Block upwards and side- 
ways, in the Z and X directions ; but in the Y direction 
he is powerless, and is compelled to represent extension 
in that direction by setting the three sets of slabs 
alongside in his plane. The second and third sets de- 
note the height and breadth of the respective walls, but 
not their depth or thickness. Now, note that the Block 
extends three inches in each of the three directions. 
The plane-being can represent two of these dimensions 
on his plane; but the unknown direction he has to 
represent by a repetition of his plane figures. The cube 
extends three indies in the Y direction. He has to use 
3 sets of slabs. 

The Block is built up arbitrarily in this manner: 
Starting from Uma Mala and going up^ we come to a' 
Brown cube, and then to a Lighk-blue cube Starting 



158 



A New Era of Thought. 



fiom Urna Mala and going right, we come to an Orange 
and a Fawn cube. Starting from Urna Mala and going 
away from us, we come to a Blue and a Buff cube. 
Now^ the plane-being represents the Brown and Orange 
cubes by squares lying next to the square which repre* 
sents Urna Mala. The Blue cube is as close as the 
Brown cube to Urna Mala, but he can find no place in 
the plane where he can place a Blue square so as to 
show this co-equal proximity of both cubes to the first 
So he is forced to put a Blue square anywhere in his 
plane and say of it : ^ This Blue square represents what 
I should arrive at, if I started from Urna Mala and 
Amoved away, that is in the Y or unknown direction.** 
Now, just as there are three cubes going up, so there 
are three going away. Hence, besides the Blue square 
placed anywhere on the plane, he must also place a Buff 
square beyond it, to show that the Block extends as far 
away as it does upwards and sideways. (Each cube 
being a different colour, there will be as many different 
colours of squares as of cubes.) It will easily be seen 
that not only the Gold square, but also the Orange and 
every other square in the first set of slabs must have two 
other squares set somewhere beyond it on the plane to 
Represent the extension of the Block away, or in the 
unknown Y direction. 

Coming now to the representation of a four-dimen« 
sional bfock, we see that we can show only three dimen- 
sions by cubic blocks^ and that the fourth can only be 
represented by repetitions of such blocks. There must 
be a certain amount of arbitrary naming and colouring. 
The colours have been chosen as now stated. Take the 
fint Block of the 8i Set We are familiar with its 
colours, and they can be found at any time by reference 
to Modd I. Now, suppose the Gold cube to represent 
what we'can see in our space of a Gold tessaract ; the 



ri 

M 



"I 



Representation of Four-Space. 159 

other cubes of Block i give the colours of the tessaracts 
which lie in the three directions X, Y, and Z from the 
Gold one. But what is the colour of the tessaract which 
lies next to the Gold in the unknown direction, W? 
Let us suppose it to be Stone in colour. Taking out 
Block 2 of the 81 Set and arranging it on the pattern of 
Model 9^ we find in it a Stone cube. But, just as there are 
three tessaracts in the X, Y, and Z directions, as shown 
by the cubes in Block i, so also must there be three 
tessaracts in the unknown direction, W. Take Block 3 
of the 81 Set This Block can be arranged on the 
pattern of Model 3. In it there is a Silver cube where 
the Gold cube lies in Block i. Hence, we may say, the 
tessaract which comes next to the Stone one in the 
unknown direction from the Gold, is of a Silver colour. 
Now, a cube in all these cases represents a tessaract 
Between the Gold and Stone cubes there is an inch in 
the unknown directioa The Gold tessaract is supposed 
to be Gold throughout in all four directions^ and so also 
is the Stone. We may imagine it in this way. Sup- 
pose the set of three tessaracts, the Gold, the Stone, and 
the Silver to move through our space at the rate of an 
inch a minute. We should first see the Gold cube 
which would last a minute, then the Stone cube for a 
minute, and lastly the Silver cube a minute. (This is 
precisely analogous to the appearance of passing cubes 
to the plane-being as successive squares lasting a 
minute.) After that, nothing would be visible. 

Now, just as we must suppose that there are three 
tessaracts proceeding from the Gold cube in the un- 
known direction, so there must be three tessaracts ex- 
tending in the unknown direction from every one of the 
27 cubes of the first Block. The Block of 27 cubes is 
not a Block of 27 tessaracts, but it represents as much* 
of them as we can see at once in our space ; and they 



i6o 



j4 New Era of TlioughL 



form the first portion or layer (IBcc the first wall of 
cubes to the plane-being) of a*set of cighty-one tessa- 
ractsi extending to equal distances in all four directions. 
Thus, to represent the whole Block of tessaracts there 
are 8i cub^ or three Blocks of 27 each. 

Now, It is obvious that, just as a cube has various 
plane boundaries, so a tcssaract has various cube bound- 
aries The cubes of the tessaract, which we have been 
regarding, have been those containing the X, Y, and Z 
directions, just as the plane-being regarded tlie Moena 
faces conUining the X and Z directions. And, as long 
as the tessaract is unchanged in its position with regard 

' to our space, we can never sec any portion of it which 
is in the unknown direction. Similarly, we saw that a 
plane-being could not see the parts of a cube which went 
in the third direction, until the cube was turned round 
one of its edges. In order to make it quite clear what 

^partsof acube*came into the plane, we gave distinct 
names to them. Thus, the squares conUining the Z and 
X directions were called Moena and Murex ; those con* 
taining the Z and Y, Alvus and Proes ; and those the 
X and Y, Syce and MeL Now, similarly with our four 
axes, any three will determine a cube. Let the tessaract 

. in its normal position have the cube Mala determined by 
the axes Z, X, Y. Let the cube Lar be that which is 
determined by X, Y, W, that is, the cube which, starting 
ifom the X Y plane, stretches one inch in the unknown 
or W directkm. Let Vesper be the cube determined by 
Z»Y,W,andPluviumby Z.X,W. And let these cubes 
have opposite cubes of the following names : 

Mala has Maigo 

Lar w Velum 

Vesper M Idas 

Plamm ,, Tela 

Anotfier way oflooking at the matter is this. When 



Representation of Four-Spfue. 161 

a cube is generated from a square, each of the lines 
bounding the square becomes a square, and the square- 
itself becomes a cube, giving two squares in its initial 
and final positions. When a cube moves in the new 
and unknown direction, each of its planes traces a cube 
and it generates a tessaract, giving in its initial and 
final positions two cubes. Thus there are eight cubes 
bounding the tessaract, six of them from the six plane 
sides and two from the cube itself. These latter two 
are Mala and Marga The cubes from the six sides are : 
Lar from Syce, Velum from Mel, Vesper from Alvus, 
Idus from Proes, Pluvium from Moena, Tela from Murex. 
And just as a plane-being can only see the squares of a 
cube, so we can only see the cubes of a tessaract It 
may be said that the cube can be pushed partly through 
' the plane, so that the plane-being sees a section between 
Moena and Murex. Similarly, the tessaract can be 
pushed through our space so that we can see a section 
between Mala and Margo. 

There is a method of approaching the matter, which 
settles all difiiculties, and provides us with a nomencla- 
ture for every part of the tessaract We have seen how 
by writing down the names of the cubes of a block, and 
then supposing that their number increases, certain sets 
of the names come to denote points, lines, planes, and 
solid. Similarly, if we write down a set of names of 
tessaracts in a block, it will be found that, when their 
number is increased, certain sets of the names come to 
denote the various parts of a tessaract 

For this purpose, let us take the 81 Set, and use the 
cubes to represent tessaracts. The whole of the 81 
cubes make one single tessaractic set extending three 
inches in each of the four directk>ns» The names must 
be remembered to denote tessaracts. Thus, Corvus is a* 
tessaract which has the tessaracts Cuspis and Nugae to 

If 



i6a 



A New Era of Thought. 



-A 



the right, Arctos and Ilex above it» Dos and Cista away 
ffom ity and Ops and Spira in the fourth or unknown 
direction from it It will be evident at once, that to 
write these names in any representative order we must 
adopt an arbitrary system. We require them running 
in four directions ; we have only two on paper. The X 
direction (from left to right) and the Y (from the bottom 
towards the top of the page) will be assumed to be truly 
represented. The Z direction will be symbolized by 
writing the names in floors, the upper floors always 
preceding Uie lower. Lastly, the fourth, or W, direction 
(which has to be s)rmbolized in three-dimensional space 
by setting the solids in an arbitrary position) will be 
signified by writing the names in blocks, the name which 
stands in any one place in any block being next in the 
W direction to that which occupies the same position in 
the block before or after it Thus, Ops is written in the 
same place in the Second Block, Spira in the Third 
Blocks as Corvus occupies in the First Block. 

Since there aie an equal number of tessaracts in each 
of the four directions, there will be three floors Z when 
there are three X and Y. Also, there will be three 
Blocks W. If there be four tessaracts in each direction, 
there will be four floors Z, and four blocks W. Thus, 
when the number in each direction is enlarged, the 
number of blocks W is equal to the number of tessaracts 
in each known direction. 

On ppi 136, 137 were given the names as used for a 
cubic block of 27 or 64. Using the same and more 
names for a tessaractic Set, and remembering that each 
name now represents, not a cube, but a tessaract, we 
obtain the following nomenclature, the order in which 
the names are written being that stated above : 



Representation of Four-Space. 163 



Third Block. 



Second Block. 



First Block. 



i! 






164 



A New Era of ThaughL 



It is evident that this set of tessaracts could be 
increased to the number of four in each direction, 
the names being used as before for the cubic blocks 
on PP> 136b I37» and in that case the Second Block 
would be duplicated to make the four blocks required 
in the unknown direction. Comparing such an 81 Set 
and 256 Set, we should find that Cuspis, which was 
a single tessaract in the 81 Set became two tessaracts 
in the 256 Set And, if we introduced a larger number, 
it would simply become longer, and not increase in 
any other dimension. Thus» Cuspis would become the 
name of an edge. Similarly, Dos would become the 
: name of an edge, and also Arctos. Ops, which is found 
in the Middle Block of the 81 Set, occurs both in the 
Second and Third Blocks of the 256 Set ; that is, it also 
tends to elongate and not extend in any other direction, 
and may therefore be used as the name of an edge of 
a tessaract 

Looking at the cubes which represent the Syce tessar- 
acts^ we find that,' though they increase in number, they 
increase only in two directions ; therefore. Syce may be 
taken to signify, a square. But, looking at what comes 
from Syce in the W direction, we find in the Middle 
Block of the 81 Set one Lar, and in the Second and 
Third Blocks of the 256 Set four Lars each. Hence, Lar 
extends in three directions, X, Y, W, and becomes a cube. 
Similarly, Moena is a plane; but Pluvium, whidi proceeds 
from it, extends not only sideways and upwards like 
Moena, but in the unknown direction alsa It occurs 
in both Middle Blocks of the 256 Set Hence, it also 
is a cube. We have now considered such parts of the 
Sets as contain on^ two, and three dimensions. But 
there is one part which contains four.. It is that named 
Tessaract In tiie 256 Set there are eight such cubes in 
the Second, and eight in the Third Block; that is, they 



-: 



ReprssiHtation of Four-Space. 165 

extend Z, X, Y, and also W. They may, therefore, be 
considered to represent that part of a tessaract or 
tessaractic Set, which is analogous to the interior of a 
cube. 

The arrangement of colours corresponding to these 
names is seen on Model i corresponding to Mala, Model 
2 to Margo, and Model 9 to the intermediate block. 

When we take the view of the tessaract with which 
we commenced, and in which Arctos goes Z, Cuspis X, 
Dos Y, and Ops W, we see Mala in our space. But 
when the tessaract is turned so that the Ops line goes 
-X, while Cuspis is turned W, the other two remaining 
as they were, then we do not see Mala, but that cube 
which, in the original position of the tessaract, contains 
the Z» Y, W, directions, that is, the Vesper cube. 

A plane-being may begin to study a block of cubes 
by their Syce squares ; but if the block be turned round 
Dos, he will have Alvus squares in his space, and he 
must then use them to represent the cubic Block. So, 
when the tessaractlc Set is turned round, Mala cubes 
leave our spacer and Vespera enter. 

There are two wa}r8 which can be followed in studying 
the Set of tessaracts. 

L Each tessaract of one inch every way can be 
supposed to be of the same colour throughout, so that, 
whichever way it be turned, whichever of its edges 
coincide with our known axes, it appears to us as a cube 
of one uniform colour. Thus^ if Urna be the tessaract, 
Urna Mala would be a Gold cube, Urna Vesper a Gold 
cube, and so on. This method is, for the most part; 
adopted in the following pages. In this cas^ a whole 
Set of 4x4x4x4 tessaracts would in coloun resemble 
a set composed of four cubes like Models i, 9, 9, and 2. 
But, when any question about a particular tessaract has 
to be settled, it is advantageous; for the sake of distinct- 



j66 



A New Era of Thought. 



1168% to suppose it coloured in its different r^ions as 
the whole set is coloured. 

IL The other plan i% to start with the cubic sides 
of tiie inch tessaract, each coloured according to the 
scheme of the Models i to a In this case, the lines, if 
shown at all, should be very thin. For, in fact, only 
the faces would be seen, as the width of the lines would 
only be equal to the thickness of our matter in the 
fourth dimension, which is indistinguishable to the 
senses. If such completely coloured cubes be used, less 
emv is likely to creep in ; but it is a disadvantage that 
each cube in the several blocks is exactly like the others 
in that block. If the reader make such a set to work 
with for a time, he will gain greatly, for the real way of 
acquiring a sense of higher sp^oe is to obtain those 
experiences of the senses exactly, which the observation 
of a four-dimensional body would give. These Models 
I'-Sare called sides of the tessaract 

To make the nuitter perfectly dear, it is best to sup- 
pose that any tessaract or set of tessaracts which we 
examine^ has a duplicate exactly similar in shape and 
arrangement of parts, but different in their colouring. 
In the first; let each tessaract have one colour through- 
out; so that all its cubes, apprehended in turn in our 
qpao^ will be of one and the same colour. In the 
duplicate let each tessaract be so coloured as to show 
its different cubic sides by thek different colours. 
Then, when we have it turned to us in different aspects, 
we shall see different cubes, and when we try to trace 
the contacts of the tessaracts with each otiber, we shall 
be helped by lealisfaig each part of every tessaract in 
its own ooloiir. 






CHAPTER VIII. 

REPRESENTATION OF FOUR-SPACE BY NAME. 
STUDY OF TESSARACTS. 

We have now surveyed all the preliminary ground, and 
can study the masses of tessaracts without obscurity. 

We require a scaffold or framework for this purpose, 
which in three dimensions will consist of eight cubic 
spaces or octants assembled round one point, as in two 
dimensions it consisted of four squares or quadrants 
round a point 

These eight octants lie between the three axes Z, X, 
Y, which intersect at the given point, and can be named 
according to their positions between the positive and 
negative directions of those axes. Thus the octant 
Z, X, Y, is that which is conUined by the positive por- 
tions of all three axes ; the octant Z, X, Y, that which 
is to the left of Z, X, Y, and between the positive parts 
of Z and Y and the negative of X. To illustrate this 
quite clearly, let us take the eight cubes— Uma, Moles, 
Plebs» Frenum, Uncus, Pallor, Bidcns, Ostrum— and 
place them in the eight octants. Let them be placed 
round the point of intersection of the axes ; Pallor 
Corvus, Plebs Ilex, etc, will be at that point Their 
positions will then be :— _. . 

Urna in the Octant 2XY 



Moles 
Plebs „ 
Frenum „ 
Uncus „ 
Pallor n 
Bidiens „ 
Ostnim »i 



ZXY 
ZXY 
ZXY 
ZXY 
ZXY 
ZX^ 
ZX7 



■•f 



i68 



A Ntw Era of Thomghi. 



The names used for the cubes, as tbejr aie before tts» 
are as follows:-. 



--J. ( Afcus Mala 

™{LawMMaIa 

'"*(A3d«MaU 

3-m-j ( Portif Mala 
"rr^^OrciuMala 
""^(VeAomMala 

„^ (TdamMala 
J!** < Poim Mala . 
"~'(CferrtiMala 



IVid 
Floor. 



•I 



An Mala 
PraadaMala 
Cortex Mala 



^^1^,, ^ i Pilum Mala 

Floor. )^^*JJ*I* 
ICardoMala 

rmc fAgmenMala 
Plo^ I Crates Mala 
* ITIiyrsus Mala 



Yijrf (MaroMala 
PtoorJSplculaMala 
V Comet Mala 

S^^^^r Ala Mala 
lS^{u»««Mala 
vOitniniMala 



Fint 
Floor* 



[Sector Mala 
I F reinm Mala 
LUmaMala 



Third Block. 

OvisMala 
Tigris Mala 
Troja Mala 

Gipeus Mala 
LacerUMala 
Loans Mala 

Nepos Mala • 
Penates Mala 
SecnrisMala 

Second Block. 

Vomer Mala 
Sacetdos Mala 
Mica Mala 

Clans MaU 

. Tessera Mala 

Cudo MaU 

LacosMala 
CuraMala 
VitU Mala 

First Block. 

MercesMala 
Mora Mala 
Tibicen Mala 

CbrtisMida 
Pallor Mala 
BidensMala 

HamaMala 
PlebsMala 
Moles Mala 



PorttoMala 
SqrnMoMala 
Aries Mala 

Tabula Mala 
Testudo Mala 
A^giusMala 

AngusUMala 
Vttlcan Mala 
Vinculum Mala 



PlumaMala 
Hydra Mala 
Flagdlum Mala 

ColusMala 
Domitor Mala 
Malleus Mala 

AnrusMala 
Limen Mala 
Scepfrum Mala 



Tyro Mala 
OUvaMala 
Vestis Mala 

AerMala 

TergnmaMala 

ScenaMala 

Remus Mala 
Sypho Mala 
SaltttsMala 



Representation of Ftmr-Space by Name. 169 

Their colours can be found by reference to the 

Models I, 9^ 2, which correspond respectively to the 

First, Second, and Third Blocks. Thus, Uma Mala is 

Gold ; Moles, Orange ; Saltus, Fawn ; Thsrrsus, Stone ; 

Cervix, Silver. The cubes whose colours are not shown 

in the Models, are Pallor Mala, Tessera Mala, and 

Lacerta Mala, which are equivalent to the interiors 

of the Model cubes, and are respectively Light-buflC 

Wooden, and Sage-green. These 81 cubes are the cubic 

sides and sections of the tessaracts of an 81 tessaractic 

Set, which measures three inches in every direction. 

We suppose it to pass through our space. Let us call 

the positive unknown direction Ana {ie^ -h W) and the 

negative unknown direction Kata (- W). Then, as the 

whole tessaract moves Kata at the rate of an inch a 

minute, we see first the First Block of 27 cubes for one 

.minute, then the Second, and lastiy the Third, each 

lasting one minute. 

Now, when the First Block stands in the normal 
position, the edges of the tessaract that run from the 
Corvus comer of Uma Mala, are : Arctos in Z, Cuspis 
in X, Dos in Y, Ops in W. Hence, we denote this 
position by the following S3rmbol :— 

Z X Y W 
a c d o 

where a stands for Arctos, c for Cuspis, d for I>o% 
and o for Ops, and the other letters for the four axes in 
space. Of e^d, o arc'the axes of the tessaract, and can 
take up diflferent directions in space with regard to us. 



Let us now take a smaller four-dimensional set 
tile 81 Set let us take the following :— . 

Z X Y W 
a e d o 



Of 



I70 



A New Era of Thought. 



' SiooND Block. 

%M:tmA FImv /^^^'^ ^^^ Tetsera Mida 

^^ \c«do MaU Cttdo Mala 

\ThyrtusMaU Vitta Mab 



Second Floor. 



Fint Floor. 



First Block. 

(Uncus Mala 
Ostrum Mala 

rFremmi Mala 
lUmaMala 



Pallor Mala 
Bidens Mida 

Plebs Mala 
Moles Mala 



« Let the First Block be put up before us in Z X Y, 
(Uma Corvus is at the junctioii of our axes Z X Y). 
The Second Block is now one inch dbtant in the un- 
known direction ; and, if we suppose the tessaractic 
Set to move through our space at the rate of one 
inch a minute^ the Second will enter in one minute, and 
replace tibe first But, instead of this, let us suppose 
the tessaracts to turn so that Ops, which now goes W. 
shall go —X. Then we can see in our space that cubic 
side of each tessaract which is contained by the lines 
Arctos, Dos, and Ops, the cube Vesper ; and we shall 
no longer have the Mala sides but the Vesper sides of 
the tessaractic Set in our space. We will now build 
it up in its Vesper view (as we built up the cubic Block 
in its Ahrus view). Take the Gold cube, which now 
means Uma Vesper, and place it on the left hand of its 
former position as Uma Mala, that is, in the octant 
Z IC Y. Thyrsus Vesper, whidi previously lay just 
beyond Uma Vesper in the unknown direction, will 
now lie just beyond it in the —X direction, that is, 
to the left of it The tessaractic Set is now in the 

poattion ,^^]^ ^ (the minus sign over the o meaning 



■f 



• 



V9$f€r$ 



Mdu 







{r#/taflr/..t7i. 



M 



Ri^nsiHlatum of Four-Space by Nanu. 171 

that Ops runs in the n^ative direction), and its Vespers 
lie in the following order :^ 



Second Floor, 



Second Block. 

Tessaim 



Fim Floor. 



LCudo 

{Cura 
Vitu 



First Block. 



Pallor 
Bidens 

Plebs 
Moles 



Uncttft 
Oitnim 

Fienvm 
Una 



The name Vesper is left out in the above list for the 
sake of brevity, but should be used in studying the 
. positions. 

On comparing the two lists of the Mala view and 
Vesper view, it will be seen that the cubes presented in 
the Vesper view are new sides of the tesisaract, and that 
the arrangement of them is different from that in the 
Mala view. (This is analogous to the changes in the 
slabs from the Moena to Alvus view of the cubic Block.) 
Of course, the Vespers of all these tessaracts are not 
visible at once in our space, any more than are the 
Moenas of all three walls of a cubic Block to a plane- 
being. But if the tessaractic Set be supposed to move 
through space in the unknown direction at the rate 
of an inch a minute, the Second Block will present 
its Vespers after the First Block has lasted a minute. 
The relative position of the Mala Block and the Vesper 
Block may be represented in our space as in the dia- 
gram. Fig. 2a But it must be distinctly remembered 
that this arrangement is quite conventional, no more 
real than a plane-being^s symbolisatioo of die Moena 



172 



A New Era of Thought. 



Wall and the Alvus Wall of the cubic Block by the 
ammgement of their Moena and Alvus faces, with the 
•oliditjr omitted, along one of his known directions. 

The Vespers of the First and Second Blocks cannot 
be in our space simultaneously, any more than the 
Moenas of all three walls in plane space. To render 
their simultaneous presence possible, the cubic or 
tessamctic Block or Set must be broken up, and its 
'parts no longer retain their relations. This fact is of 
supreme importance in considering higher space End- 
less fallacies creep in as soon as it is forgotten that the 
cubes are merety representative as the slabs were, and 
t^ positions in our space merely conventional and 
symbolical, like those of the slabs along the plane. 
And these fallacies are so much fostered by again sym- 
« bolizing the cubic s)rmbols and their symbolical positions 
in perspective drawings or diagrams, that the reader 
should surrender all hope of learning space from this 
book or the drawings alone, and work every thought 
out with the cubes themselves. 

If we want to see what each individual cube of the 
tessaractic faces presented to us In the last example is 
like^ we have only to consider each of the Malas simi- 
lar in its parts to Model i, and each of the Vespers to 
Model 5« And it must alwa}rs be remembered that the 
cabes^ though used to represent both Mala and Vesper 
fSices of the tessaract, mean as great a difference as the 
slabs used for the Moena and Alvus faces of .the cube. 

If the tessaractic Set move Kata through our space, 
when the Vesper faces are presented to us, we see the 
f<41owing parts of the tessaract Urna (and, therefore, 
also tiie same parts of the other tessaracts) : 

(t) Urna Vesper, which is Model 5* 

(3) A parallel section between Urna Vesper and Urna 
Idiis» which is Model II. 



r-. 



\^ 



; ■ 
1"^ 



Representation of Four-Space ky Name. 173 

(3) Urna Idus, which is Model 6. 

When Urna Idus has passed Kata our space. Moles 
Vesper enters it \ then a section between Moles Vesper 
and Moles Idus, and then Moles Idus. Here we have 
evidently observed the tessaract more minutely; as it 
passes Kata through our spacer starting on its Vesper 
side, we have seen the parts which Would be generated 
by Vesper moving along Cuspis— that is Ana. 

Two other arrangements of the tessaracts have to be 
learnt besides those from the Mala and Vesper aspect 
One of them is the Pluvium aspect Build up the Set 
in ZXlT, letting Arctos run Z, Cuspis X, and Ops Y. 
In the common plane Moena, Urna Pluvium coincides 
with Urna Mala, though they cannot be in our space 
together; so too Moles Pluvium with Moles Mala, 
Ostrum Pluvium with Ostrum Mala. And lying towards 
us, or 7, is now that tessaract which before lay in the 
W direction from Urna, viz., Thyrsus. The order will 
therefore be the following (a star denotes the cube 
whose comer is at point of intersection of the axes, and 
the name Pluvium must be understood to follow each 
of the names): 

Z X Y W 
a c e d 

SECOND Block. 

rFrenum Plebt 

\CnUes Cura 



Fim Floor. 



First Block. 



Ostnun 
Caido 



Second Floor.f 



^«»^"«*- r?siu. 



Bideai 
Cudo 

Motes 
Vitia 






»74 



A New Era of Thought. 



Thus the wait of cubes in contact with that wall of the 
Hata position which contains the Urna, Moles, Ostrum, 
and Bidens Malas, is a wall composed of the Pluviums of 
Uma, Moles, Ostium, and Bidens. The wall next to 
this, and nearer to us, is of Thyrsus^ Vtttay Cardo, Cudo, 
Pluviums. The Second Block is one inch out of our 
Space, and only enters it if the Block moves Kata. 
Model 7 shows the Pluvium cube ; and each of the cubes 
of the tessaracts seen in the Pluvium position is a Pluv- 
ium. If the tessaractic Set moved Kata, we woi^ld see 
the Section between Pluvium and Tela for all but a 
minute ; and then Tela would enter our space, and the 
Tela of each tessaract would be seen. Model 12 shows 
the Section from Pluvium to Tela. Model 8 is Tela. 
Tela only lasts for a flash, as it has only the minutest 
magnitude in the unknown or Ana direction. Then, 
Frenum Pluvium takes the place of Uma Tela ; and, 
when it passes through, we see a similar section between 
Frenum Pluvium and Frenum Tela, and lastly Frenum 
Tela. Then the tessaractic Set passes out, or Kata, our 
space. A similar process takes place with every other 
tessaract, when the Set of tessaracts moves through our 
space. 

There is still one more arrangement to be learnt If 
the line of the tessaract, which in Jhe Mala position goes 
Ana, or W, be changed into the Z or downwards direc- 
tion, the tessaract will then appear in our space below the 
Mala position ; and the side presented to us will not be 
Mala, but that nMch contains the lines Dos, Cuspis, and 
Ops. This skle is Model 3, and is called Lar. Under- 
acith the place which was occupied by Uma Mala, will 
come Uma Lar ; under the place of Moles Mala, Moles 
Lar; under the place of Frenum Mala, Frenum Lar.' 
The tessaract; which in the Mala position was an inch 
out of our space Ana, or W, from Uma Mala, will now 



)\ 



t: 



R0^res09UaiwK of Four-Spaa by Name. 175 

come into it an inch downwards, or Z, below Uma 
Mala, with its Lar presented to us ; that is. Thyrsus 
Lar will be below Uma Lar. In the whole arrange- 
ment of them written below, the highest floors are 
written first, for now they stretch downwards instead of 
upwards. The name Lar is understood after each. 

Z X Y W 
d c d a 



Second Block. 



Second Floor. 



••( 



First Floor. 



Uncus 
Ostnim 

{Ocrea 
Caitlo 



Pallor 
Bidens 

Tessera 
Cndo 



Second Floor. 



First Block. 

Frenum Plebs 



L*Uma 
Crates 



First Floor. { Thymus 



Moles 

Cura 
Vitta 



Here it is evident that what was the lower floor of 
Malas, Uma, Moles, Plebs, Frenum, now appears as if 
carried downwards instead of upwards, Lars being pre- 
sented in our q^ace instead of Malas. This Block of 
Lars is what we see of the tessaract. Set when the 
Arctos line, which in the Mala position goes up^ is 
turned into the Ana, or W, direction, and the Ops line 
comes in downwards. * 

The rest of the tessaracts, which consists of the cubes 
opposite to the four treated above, and of the tessaractic 
space between them, is all Ana our space. If the tessar- 
act be moved through our space — for instance, when the 
Lars are present in it— we see» taking Uma alone^ first 
the section between Uma Lar and Uma Velum (Model 



176 



A, New Era of Thought. 



loX and then Uma Velum (Model 4), and similarly the 
lections and Veltims of each tessaract In the Set When 
the First Block has passed Kata our space, Ostrum 
Lar enters ; and the Lars of the Second Block of tessa- 
ncts occupy the places just vacated by the Velums of 
the First Block. Then» as the'tessaractic Set moves 
00 Kata» the sections between Velums and Lars of the 
Second Btock of tessarscts enter our space, and finally 
tfadrVeluaiSL Then the wh<4e tessaractic Set disappears 
finom our space. 

When we have learnt all these aspects and passages^ 
we have experienced some of the principal features of 
tjiis small Set of tessanicts. 



I.: 






s 



CHAPTER \y^ 

FURTHER STUDY OF TESSARACTS. 

Whex the arrangement of a small set has been 
mastered, the different views of the whole 81 Set should 
be learnt It is now clear to us that, in the list of the 
names of the eighty-one tessaracts given above, those 
which lie in the W direction appear in different blocks^ 
while those that lie in the Z, X,or Y directions can be 
found in the same block. Therefore, from the arrange- 

Z X Y W 
-ment given, which is denoted W^cdo*^ "^^^ 

briefly by a cdo^ we can form any other arrangement 

To confirm the meaning of the symbol acdo for 
position, let us remember that the order of the axes 
known in our space will invariably be Z X Y, and the 
unknown direction will be sUted last, thus : Z X Y W. 
Hence, if we write addc^ we know that the position or 
aspect intended is that in which Arctos (tf) goes Z, Ops 
(4) nq;ative X, Dos (d) Y, and Cuspis (^) W. And such 
an arrangement can be made by shifting the nine cubes 
on the left side of the First Block of .the eighty-one tes- 
saracts, and putting them into the Z X Y octant, so that 
they just touch their former position. Next to them, to 
their left, we set the nine of the left side of the Second 
Block of the 81 Set ; and next to these again, on their 
left, the nine of the left side of the Third Block. This 
Block of twenty*seven now represents Vesper Cube% 
which have only one square identical with tlie Mala 

Iff V 



178 



A New Era of Thought. 



cubes of the. previous blocks^ from which they were 

Similarly the Block which is one inch Ana, can be 
made by taking the nine cubes which come vertically 
in the middle of each of the Blocks in the first position, 
and arranging them in a similar manner. Lastly, the 
Block which lies two inches Ana, can be made by taking 
the right sides of nine cubes each from each of the three 
original Blocks, and arranging them so that those in the 
Second original Block go to the left of those in the First, 
and those in the Third to their left. In thb manner we 
should obtain three new Blocks, which represent what 
we can see of the tessaracts, when the direction in which 
Uma, Moles, Saltus lie in the original Set, is turned W. 

The Pluvium Block we can make by taking the front 
wall of each original Block', and setting each an inch 
nearer to us, that is ^ Y. The far sides of these cubes 
afe Moenas of Pluviums. By continuing this treatment 
of the other walls of the three original Blocks parallel to 
the front wall, we obtain two other Blocks of tessaracts. 
The three together are the tessaractic position aedd^ for 
in all of them Ops lies in the -- Y direction, and Dos 
has been turned W. 

The Lar position is more difficult to construct To 
l^ut the Lars of the Blocks in their natural position in 
our space, we must start with the original Mala Blocks, 
at least three inches above the table. The First Lar 
Block is made by taking the lowest floors of the three 
Mala Blocks, and placing them so that that of the 
Second is below that of the First, and that of the Third 
below that of the Second. The floor of cubes whose 
dii^^al runs from Uma Lar to Remus Lar, will be at 
the top of the Block of Lars ; and that whose diagonal 
goes from Cervix Lar to Angusta Lar, will be at the 
bottom. ' The next Block of Lars will be made by 



Further Study of Tessaracts. 179 

taking the middle horizontal floors of the three original 
Blocks, and placing them in a similar succession— the 
floor from Ostrum Lar to Aer Lar being at the top, that 
from Cardo Lar to Colus Lar in the middle, and VerbuDi 
Lar to Tabula Lar at the bottom. The Third Lar 
Block is composed of the top floor of the First Block on 
the top— that is, of Comes Lar to Tyro Lar, of Cortex 
Lar to Pluma Lsr in the middle^ and Axis Lar to Portio 
Lar at the bottom. 



CHAPTER X. 

CYCLICAL PROJECTIONS. 

Let us denote the original position of the cube» that 
whefdn Arctos goes Z, Cuspb X, and Dos Y, by the 
expression, 

ZXY 
«: a c d (i) 

If the cube be turned round Cuspis, Dos goes Z, 
Cuspis remains unchanged, and Arctos goes Y, and we 
have the position, 

ZXY 

• ' d c a 

2 
where ^ means that Dos runs in the n^ative direction 

of the Z axis from the point where the axes intersect 

^ Z 

We might write ^ but it is preferable to write ^ 

If we next turn the cube round the line, which runs 
Y, that is^ round Arctos, we obtain the position, 

ZXY 

c d a (2) 

and hy means of this double turn we have put rand d 
in the places of a and c Moreover, we have no nega- 
tive directiona. This position we call simply r d n. 
If from it we turn the cube round a, which runs Y, 

ZXY 

weget ^ - ^* and if, then, we turn it round Dos we get 

m C 4^ 

ZXY 

^ ^ * or simply ^« c . This last is another position in 



Cyclical Projections. 



181 



which all the lines are positive, and the projections, in- 
stead of lying in different quadrants, will be contained 
in one. 

The arrangement of cubes in acd y»t know. That 
{ncdahi 



Third (^^**'« 



Oliva Tyro 

Tei^g^m Aer 

Sypho Remus 



Second 
Floor. 



I Ttbicen 
Bidens 
L Moles 



Mora 
Pallor 
Plebs 



Meroes 

Cortis 

Hama 



(Comes 
!^!'»^ \ Ostrum 
^'«>''- I Uma 



Spicula Mars 

Uncus Ala 

Frenum Sector 



It will be found tliat learning the cubes in this position 
gives a great advantage, for thereby the axes of the cube 
become dissociated with particular directions in space. 

Thtdac position gives the following arrangement: 



III 


Aer 
Coitit 

Ala 


Tyro 

Merces 

Mars 


Sypho 

Plebs 

Frenum 


Ter^^m 

Pallor 

Uncus 


Oltva 
Mora 
Spicula 


III 


Scena 

Bidens 

Ostrum 


Vestis 

Tibicen 

Comes 



The sides, which touch the vertical plane in the first 
position, are respectively, in acd Moena, in cd a Syce, 
in^a^Alvus. 

Take the shape Urna, Ostrum, Moles, Saltus, Scena, 
Sypho, Remus, Aer, Tyra This gives in a c d tiie 
projection : Uma Moena, Ostrum Moena, Moles Moena, 



I82 



A New Era of Thoug/iL 



Saltus Moena* Scena Moena, Vestis Moena. (If the 
diffefent positions of the cube are not well known, it b 
best to have a list of the names before one, but in every 
case the block should also be built, as well as the names 
used.) The same shape in the position r ^ a is, of course^ 
expressed in the same words, but it has a different ap- 
pearance. The front face consists of the Syces of 



Saints 


Sypho 


Remus 


Moles 


PJebs 


Hama 


Uma 


Frenum 


Sector 



And taking the. shape we find we have Uma, and we 
know that Ostrum lies behind Uma, and does not come 
M; next we have Moles, Saltus^ and we know that 
Scena lies behind Saltus and does not come in ; lastly, 
we have Sypho and Remus, and we know that Aer and 
Tyro are in the Y direction from Remus, and so do not 
come in. ^ Hence, altogether the projection will consist 
only of the Syces of Uma, Moles, Saltus, Sypho^ and 
Remus. 

Next, taking the position dfn r, the cubes in the front 
Cue have their Alvus sides against the plane, and are : 



Sector 


Ala 


Mars 


Ftenum 


Ubom 


Spicula 


Uma 


OttruRi 


Comes 



And, taking the shape, we find Uma, Ostnim ; Moles 
aiul Saltus are hidden by Uma, Scena is behind Ostrum« 
Sypho gives Frenum, Remus gives Sector, Aer gives Ala, 
and Tyro gives VLwn. All these are Alvus sidesL 

Let us now take the reverse problem, and, given the 
three cyclical projections, determine the shape. Let 
the « ^ ^ projectkxi be the Moenas of Uma, Ostram, 
Kdens, Scena, Vestis. Let the c ^ « be the Syces of 
UrDa,Freniiai, Plebs»iSypha^and the ^^^ be the Alvus 
of UiM, FreBum* Uncus^ Spkula. Now, from m<dwt 






Cyclical Projections. 



183 



have Uma, Frenum, Sector, Ostram, Uncus, Ala, Bidens, 
Pallor, Cortis, Scena, Tergum, Aer, Vestis, Oliva, Tyra 
Fiom c d a xtt have Uma, Ostrum, Comes, Frenum, 
Uncus^ Spicula, Plebs, Pallor, Mora, Sypho» Tergum, 
Oliva. In order to see how these will modify each 
other, let us consider ^ta cd solution as if it were a 
set of cubes in the r ^ a arrangement Here, those that 
go in the Arctos direction, go away from the plane of 
projection, and must be represented by the Syce of the 
cube in conUct with the plane. Looking at the a r </ 
solution we writedown (keeping those together which go 
away from the plane of projection) : Urna and Ostrum, 
Frenum and Uncus, Sector and Ala, Bidens, Pallor, 
Cortis, Scena and Vestis, Tergum and Oliva, Aer and 
Tyra Here we see that the whole cda face -is filled up 
in the projection, as far as this solution is concerned. 
But in ih^cda solution we have only Syces of Urna, 
' Frenum, Plebs, Sypha These Syces only indicate the 
presence of a cerUin number of the cubes stated above 
as possible from the Moena projection, and those are 
Uma, Ostrum, Frenum, Uncus, Pallor, Tergum, Oliva. 
This is the result of a comparison of the Moena pro- 
jection with the Syce projectk>n. Now, writing these 
last named as they come in the 1/ a ^ projection, we 
have Uma, Ostram, Frenum, Uncus and Pallor and 
Tergum, Oliva. And of these Ostram Alvus is wanting 
in the 1/ a ^ projection u given above. Hence Ostram 
wiU be wanting in the final shapes and we have as the 
final solution: Uma^ Frenum, Uncus, PaU6r, Tergum, 
Oliva. 



CHAPTER XL 

A TESSARACnC FIGURE AND ITS PROJECTIONS. 

We will now consider a fourth-dimensional shape com* 
posed of tessaracts» and the manner in which we can 
obtain a conception of it The operation is precisely 
analogous to that described in chapter VL, by which a 
plane being could obtain a conception of solid shapea 
It is only a little more difficult in that we have to deal 
with one dimension or direction more, and can only do 
96 symbolically. 

We will assume the shape- to consist of a certain 
number of the 8i tessaracts» whose names we have 
given on p. i68. Let it consist of the thirteen tessaracts: 
Uma, Moles, Plebs, Frenum, Pallor, .Tessera, Cudo, Vitta, 
Cuni» Penates, Polus, Orcus, Lacerta. 

Firstly, we will consider what appearances or projec* 
tibns these tessaracts will present to us according as the 
tessaractic set touches our space with its (a) Mala cubes, 
(d) Vesper cubes, (c) Pluvium cubes, or (d) Lar cubes. 
Secondly, we will treat the converse question, how the 
shape can be determined when the projections in each 
of those views are given. 

Let us build up in cubes the four different arrange- 
ments of the tessaracts according as they enter our space 
on their Mala, Vesper, Pluvium or Lar sides. They can 
only be printed by symbolizing two of the directions. 
In the folKnring Ubulations the directions Y, X will at 



A Tessaractic Figure and its Prqfectums. 185 

once be understood. The direction Z (expressed by the 
wavy line) indicates that the floors of nine, each printed 
nearer the top of the page, lie above those printed nearer 
the bottom of it The direction W is indicated by the 
dotted line, which shows that the floors of nine lying to 
the left or right are in the W direction (Ana) or the — W 
direction (Kata) from those which lie to the right or 
left. For instance, in the arrangement of the tessaracts, 
as Malas (Table A) the tessaract Tessara, which b 
exactly in the middle of the eighty-one tessaracts has 



Domitor on its right side 


or in the 


X direction. 


Ocrea on its left „ 


»» 


-X , 


Glans away from us ^ 


H 


Y „ 


Cudo nearer to us „ 


n 


-Y „ 


Sacerdos above it „ 


» 


z „ 


Cura below it „ 


n 


-z „ 


Lacerta in the Ana or 




w , 



Pallor in the Kata or 



-W 



Similarly Cervix lies in the Ana or W direction from 
Uma, with Thyrsus between them. And to take one 
more instance, a journey from Saltus to Arcus would 
be made by travelling Y to Remus, thence - X to Sector, 
thence Z to Mars, and finally W to Arcus. A line from 
Saltus to Arcus is therefore a diagonal of the set of 
81 tessaracts, because the full length of its side has 
been traversed in each of the four directions to reach 
one from the other, i>. Saltus to Remus^ Remus to 
Sector, Sector to Mars, Mars to Arcus. 

The relation between the four different arrangements 
shown in the tables A, B, C, and D, will be understood 
from what has been said in chapter VIII. about a small 
set of sixteen tessaracts. A glance at the lines, which 
indicate the directions in each, will show the changes 




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I90 



A Nm Era of Thought. 



effected by turning the tessaracts fiom the Mala presen- 
tetion. 

In the Vesper presentation : 
The tessaracts — 
(eg. Uma, Ostrum^ Comes), which ran Z still run 2L 
(eg. Uma, Frenum, Sector), ^ Y „ Y. 

(^. Uma, Mole% Saltus), „ X now run W. 

{fg. Uma, Thyrsiis, Cervix), » W ^ •X. 

In the Pluvium presentation : 
The tessaracts— . 
(r^. Uma, Ostrum, Comes), which i 
^€g. Uma, Moles, Saltus), „ 

{fg. Uma, Frenum, Sector), ^ 
(tg. Uma, Thyrsusi Cervix), ^ 



Z still run Z. 
X „ X 
Y now run W. 
W . -Y. 



In the Lar presentation : 
The tessaracts — 

(eg. Uma, Moles, Saltus), which ran X still run X 

(eg. Uma, Frenum, Sector), „ Y „ Y. 

(4g. Uma, Ostrum, Comes), ^ Z now run W. 

(r/; Uma, Thyrsus, Cervix), ». W „ -Z. 

This relation was already treated in chapter IX., but 

it is well to have it veiy clear for our present purpose. 

For it is the apparent change of the relative positions 

'* of the tessaracts in each presentation, which enables us 

to determine any body of thenk 

In considering the projections, we always suppose our- 
selves to be situated Ana or W towards the tessaracts, 
and any movement to be Kata or -W through our 
space. For instance, in the Mala presentation we have 
first in our space the Malas of that block of tessaracts, 
mdiich Is the last in the -^ W direction. Thus, the Mala 
projection of any given tessaract of the set is that Mala 



\ 



1 



ri 



A Tcssaractic Figure and Us Projections. 191 

in the extreme -W block, whose place its (the given 
tessaract's) Mala would occupy, if the tessaractic set 
moved Kata until the given tessaract reached our space. 
Or, in other words, if all the tessaracts were transparent 
except those which constitute the body under considera- 
tion, and if a light shone through Four-space from the 
Ana (W) side to the KaU (- W) side, there would be 
darkness in each of those Malas, which would be occu- 
pied by the Mala of any opaque tessaract, if the tes- 
saractic set moved Kata. 

Let us look at the set of 81 tessaracts we have built 
up in the Mala arrangements, and trace the projections 
in the extreme - W block of the thirteen of our shape. 
The latter are printed in italics in Table A, and their 
projections are marked {. 

Thus the cube Uncus Mala b tlie projection of the 
tessaract Orcus, Pallor Mala of Pallor and Tessera and 
Tacerta, Bidens Mala of Cudo, Frenum Mala of Frenum 
'and Polus, Plebs Mala of Plebs and Cura and Penates, 
Moles Mala of Moles and Vitta, Urna Mala of Uma. 

Similarly, we can trace the Vesper projections (Table 
B). Orcus Vesper is the projection of the tessaracts 
Orcus and Lacerta, Ocrea Vesper of Tessera, Uncus 
Vesper of Pallor, Cardo Vesper of Cudo, Polus Vesper 
of Polus and Penates, Crates Vesper of Cura, Frenum 
Vesper of Frenum and Plebs, Urna Vesper of Uma and 
Moles, Thyrsus Vesper of Vitta. Next in the Pluvium 
presentation (Table C) we find that Bidens Pluvium is 
the projection of the tessaract Pallor, Cudo Pluvium of 
Cudo and Tessera, Luctus Pluviuni of Lacerta, Verbum 
Pluvium of Orcus, Uma Pluvium of Urna and Frenum, 
Moles Pluvium of Moles and Plebs, Vitta Pluvium of 
Vitta and Cura, Securis Pluvium of Penates, Cervix 
Pluvium of Polus. Lastiy, in the Lar presentation 
(Table D) we observe that Frenum Lar is the projection 



192 A New Era of Thought. 

of Frnium, Piebs Lar of Plebs and Pallor, Moles Lar 
of Moks, Uma Lar of Urna, Cura Lar of Cura and 
Tessara, Vitu I-ar of Vitto and Cudo, Penates Lar of 
Penates and Lacerta, Poliir Lar of Polus and Orcus. 

Secondly, we will treat the converse problem, how to 
determine the shape when the projections in each pre- 
sentatlon are given. Looking back at the list just given 
above, let us write down in each presenUtk>n the pro* 
jections only. 

Mala projections: 
Uncus, Pallor, Bidens, Frenum, Plebs, Moles, Uma. 

V^per projections : 
Orcu% Ocrea, Uncus, Cardo^ Polus, Crates, Frenum, 
Uma, Thyrsus. 
Pluvium projections : 
Bidens, Cudo, Luctus, Verbum, Uma, Moles, Vitta, 
Securis, Cervix. 
Lar projections : 
Frenum, Plebs, Moles, Uma, Cura, Vitta, Polus, 
Penates. 

Now let us determine the shape indicated by these 
projections. In now using the same tables we must not 
notice the italics, as the shape is supposed to be un- 
known. It is assumed that the reader is building the 
' problem in cubes. From the Mala projections we might 
infer the presence of all or any of the tessaracts written 
in the brackets in tlie following list of the Mala presen- 
tation. 

(Uncus, Ocrea, Orcus) ; (Palbr, Tessera, LacerU) ; 

(Bidens, Cudo, Luctus) ; (Frenum, Crates, Polus) ; 

(Plebs, Cura, Penates) ; (Moles, Vitta, Securis) ; 

(Uma, Thyrsus, Cervix). 

Let us suppose them all to be present in our shape, 



^ 



M 






A Tessaractic Figure and Us ProjedioHS. 193 

and observe what their appearance would be in the 
Vesper presentation. We mark them all with an asterisk 
in Table B. In addition to those already marked we 
must mark (f) Verbum, Cardo, Ostrum, and then we 
see all the Vesper projections, which would be formed 
by all the tessaracts possible from the Mala projections. 
Let us compare these Vesper projections, vis. Orcus, 
Ociea, Uncus, Verbum, Cardo, Ostram, Polus, Crates, 
Frenum, Cervix, Thyrsus, Uma, with the given Vesper 
projections. We see at once that Verbum, Ostmm, and 
Cervix are absent Therefore, we may conclude that 
all the tessaracts, which would be implied as possible by 
their presence, are absent, and of the Mala possibili- 
ties may exclude the tessaracts Bidens, Luctus, Securis, 
and Cervix itself. Thus, of the 21 tessaracts possible 
in the Mala view, there remain only 17 possible, both 
irf the Mala and Vesper views, vis. Uncus, Ocrea, 
Orcus, Pallor, Tessera, Lacerta, Cudo^ Frenum, Crates, 
Polus, Plebs, Cura, Penate% Moles, Vitta, Uma, Thyrsus. 
This we call the Mala- Vesper solution. 

Next let us take the Pluvium presentatioa We again 
mark with an asterisk in Table C the possibilities in- 
ferred from the Mala- Vesper solution, and Ufce the 
projections those possibilities would produce. The ad- 
ditional projections are again marked (t). There are 
twelve Pluvium projections altogether, viz. Bidens, Os- 
trum, Cudo, Cardo, Luctus, Verbum, Uma, Mole% Vitta, 
Thyrsus, Securis, Cervix ' Again we compare these with 
the given Pluvium projections, and find three are absent; 
viz. Ostmm, Cardo, Thyrsus. Hence the tessaracts 
implied by Ostrum and C^ardo and Thyrsus cannot be 
in our shape, viz. Uncus, Ocrea, Crates, nor Thyrsus 
itselC Excluding these four from the seventeen possi- 
bilities of the Mala-Vesper solution we have left the 
thirteen tessaracU: Orcus, Pallor, Tessera, Lacerta, Cudo^ 

O 



194 



A Niw Era of Thought. 



Frenutn, Polus, PletM» Cura, Penates, Moles, Vitta, Uma. 
This we call the Mala-Vesper-Pluvium solutioik 

Lastly, we have to consider whether these thirteen 
tessaracts are consistent with the given Lar projections. 
We mark them again on Table D with an asterisk, and 
we 'find that the projections are exactly those given, viz. 
Frenum, Plebs, Moles, Urna, Cura, Vitta, Polus, Penates. 
Therefore^ we have not to exclude any of the thirteen, 
and can infer that they constitute the shape, which 
produces the four different given views or projections. 

In fine, any shape in space consists of the possibilities 
common to the projections of its parts upon the boun- 
daries of that space, whatever be the number of its 
dimensions. Hence the simple rule for the determina- 
tion of the shape would be to write down all the* possi- 
Ulities of the sets of projections^ and then cancel all 
those possibilities which are not common to all But 
the process adopted above is much preferable^ as through 
it we may realize the gradual delimitation of the shape 
view by view. For once more we must remind ourselves 
that our great object is, not to arrive at results hf 
symbolical operations, but to realize those results piece 
by piece through realized processes. 



APPENDICE& 



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198 



A New Era of Thought. 



APPENDIX B. 

The foUmmg lisl of namcf it used to denote cabic spaces. It 
makes a cnbic block of six floors, the highest being the sixth. 

. Pons Plectrum Vulnus Arena Mensa Terminus 

I Testa Plausos Uva 0>Uis Coma Nebuk 

S Copia Comu . Solum . Munus Rixum Vitrum 

^ Ars Pervor Thyma . Colubra Seges Cor 

^ Lupus Classis Modus Plamma Mens Incoia 

Thalamus Hasta Cahunus Crinls Auriga Vallum 

. Uoteum Pinnb Puppis NupCia Aegis Cithara 

I TriumphnsCurris Lux Portus Latus Punis 

^ Regmim Fasds ^ Bellum Capellus Arbor Custos 

•SSagitU Puer ' Stella Saxum Humor Pontus 

^Nomen Imago Lapsus Quercus Mundus Proelium 

Palaestra Nuncius Bos Phareira Pumex Tibia . 






Focus 


Oram 


Fades 


Onus 


Sol 


Ofl 


Signum 


Umbra 


Opus 


Honor 


Equus 


Miles 


Tabema 


Tunis 


ChoruS' 


Lnna 


Hiems 


Ver 


Ignis 


Aether 


Pectus 


Pelagiu 


Dux 


Ventus 


Hostis 


Fignm 


Augur 


Lodns 


Fcvrum 


Pi«ii» 


Unda 


Res 


Ulmat 


Sedes 


UrtM 


GCDS 



Alea Vox 

Gelu Flumen 
Brachium Jaculum 

Corona Socius 

Rivus Imber 

Lyra Tunica 

Domus Vinum 

Lucus Agna 

Arator Pratum 

Penna Labor 

Fretum Gradus 

Finis Robur 



. Nemus Sidus 

I Venator Cenra 

^ Membfum Aqua 

Fhictus Rus 

Tofte Agcr 



I 



Lucrum 

SUva 

Anna 

Tempus 

Campus 

Cursus 

Nox 

Flos 

Carina 

Cohors 

Nolus 

Navis 

Ales Coelum Aura 

Chunor Galea . Pes 

Res Carmen Nubes 

Templum Ripa Amnis 
Columba Aequor Dama 

Moostrum Pecus Mons 

Vertex Nu Grando Arx 

Aper Plagua Hedera Frons 

Caput Castoum Lituus Tuba 

Rads ' Amphora Pars Dies 

Trabf Myrtus Fibra Nauu 

Mcta Rota Pahna Terra , 









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tai^iiii^iiii^Mi 
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303 A New Era of Thought. 



APPENDIX D. 

The following Kst gives the colours, and the yarious uses for 
them. They have already been vsed in the foregoing pages to 
distingoish the various regions of the Tessaract, and the ditterent 
individnal cubes orTessaracts hi a block. The other use suggested 
in the last cduom of the list has not been discussed; but it is be* 
lieved that it nay afford great aid to the nimd in amasstnff, 
bandlbg, and retaining the quantities of fonnube requisite in 
scientific timinbg and work. 



Bhu*k 

White 

Vermilion 

Oruige 

Light-yellow 



Rtgimrf Tlusaraeiim 
TusaracL Si Sti. 



SymboL 



Bright-blue 

Light-grey 

Indkn-red 

YeUow-ochie 

Buff 

Wood 

Brown-green 

Sage-green 

Reddish 

Chocolate 

French-giey 

Brown 

Dark-slate 

Dun 



23- 

Alvus 

Cuspis 

Murex 

Lappa 

Iter 

Lares 

Crux 

Sal 

Cista 



Plebs 

Mora 

Uncus 

Moles 

Cortis 

Penates 

OUva 

Tigris 

Orctts 

Testudo 

Sector 



Tessaract Tessara 
Tholus Troja 



MargO' 

Callis 

Velum 

Far 

Arctos 

Daps 

Portica 



Orange-vcimiiion Talus 



Stone 
' ' Qttaker>green 
Xyf^den 
Dull-green 
Indigo 
Dull-blue 
Dark-purple 
Pale-pink 
Daik-bkie 
Earthen 
Bhie 

TerracotU 
Oak 
YeUow 



Ops 
Felis 
SemiU 
Mappa 

Pagus 

Mensura 

Vena 

Moena 

Mugil 

Dee 

Cms 

Idtts 

Pagina 



Lacerta 
Tsbicen 
Sacerdos 
Scena 
Ostnun 
Aer 
Qipeus 
Portio 
Thyrsus 
Axis 
Merces 
Vulcan 
' Postis 
Verbum 
Nepos 
TabuU 
Bidens 
Augusta 
Plenum 
Remus 
Domitor 
Caido 



o 
I 
a 
3 

4 



I 

9 



+ (plus) 
— (minus) 
db ^lus or minus) 
X (multiplied by) 
+ (divided by) 
8 (equal to) 
zXi (not equal to) 
> (greater than) 
< (less than) 
: f signs 
:: ( of proportioii 
* (decimal point) 
L. (factorial) 

IfS^TpLallel) 

J (90*) (^ ^^ Angles) 

iQg.base 10 

sin. (sine) 

cos. (cosine) 

tan. (tangent) 

•0 (infinity) 

a 

b 

c 

d 



^ 



Gceen . 

Rose 

Emeiaki 

Red 

Seargreen 

Salmon 

Pale-yellow 

Porpte-brown 

Deep-crimson 

Blue-green 

Light-brown 

Deep-blue 

BricK-red 

Magenta 

Green-grey 

Light-red 

Asure 

Pale-green 

Blue-tint 

Yellow-green 

Deep-green 



Li^ht-blue 

Cnmson 

Odue 

Purple 

Leafogreen 

Turquoise 

Dark-grey 

Fawn 

Smoke 

Light-buff 

Dull-purple 

Rich-red 

Green-blue 

Burnt-sienna 

Sea-blue 

Peacock-blue 

Deep-brown 

Dark-pink 

Dark 

Dark-stone 

Silver 

Gold 

5««p-yellow 

I>m-green 



Appendix. 

Region of Ttssaraciin 
Touamei. 81 3*$. 

Budna 

Olla 

Orsa 

Olus 

Libera 

Tela 

Livor 

Opex 



303 



Proes 

Lua 

Lama 

Ur 

Offex 

Cados 

Croeta 

Lotus 

Vesper 

Panax 

Pactum 

Mango 

Lis 

Ilex 

Bolus 

Limbus 

Solia 

Luca 

Ancilla 

Orca 

Nugae 



Mala 

Sors 

Lucu 

Pator 

Silex 

Lorica 

Passer 

Meatus 

Onager 

Lensa 

Pluvium 

Spira 

Corvus . 

Via 

Cator 



SjtmM. 



Ala 
Limen 


e 
f 


Ara 


f 


Mars 


Pluma 


1 


Clans 


I 


Ovis 


Polus 


1 


Pilum 


, m 


Tergum 


n 


Crates 





Tyro 


p 


Cura 


q 


Arvus 


r 


Hama 


s 


Praeda 


t 


Vitta 


u 


Ocrea 


r 


Telum 


w 


Malleus 


X 


Vomer 


y 


Agmen 


s 


Comes 


m 


Sypho 


fi 


Mica 


I 


Arcus 


Securis 


f 


Vinculum 


f 


Colus 


J 


Saltus 


Sceptrum 


« 


Pallor 


c 


Vestis 


X 


Cortex 


H 


FlageUum 


p ' 


Luctus 


( 


Lacus 


• 


Aries 


If 


Hydra 


p 


Anguis 


9 


Laurus 


T 


Cudo 


V 


Cervix 


> 


Uma 


J 


Spicda 



J04 



A New Era of Thought. 



Appendix. 



205 



APPENDIX E. 

A TmORBlf IN FOUR-0PACB. 

Ir a pyiamid on a triaagvlar base be cat b)r a ?!•»• which {lasiei 
thm^ the three ndes of the pyramid in such mannerthat the sides 
«r the sectional triangle are not paiaUei to the corresponding sides 
ofthetrlan^ofthebase; then the sides of these two triangles, ii 
produced in pairs, wOl meet in three points which are in a straight 
Hac^ namelrt the line of faitenectioB of the sectional phme and the 
plane of the base. 

LetABCDbeapyramidon a triangnhur base A B C, and let 
abche a section each that AB,B C, AQare respectively not 
parallel to a b^ be, a c It most be understood that a is a point 
on A D, b is a point on B D, and c apoint on C D. Let, A B and 
a \ produced, meet in m. B C and b c, produced, meet in n ; and 
A C and a c, produced, meet in o. These three points, m, n, s^ 
are in the line of Intersection of the two phmes ABC and a b c 

Now, let the line a b be projected on to the plane of the base, by 
drewing Unes from a and b at right angles to the base, and meediv 
ItiBa'V; the Ifaie a' K, produced, will meet A B produced b m. 
If the Unes b c and a c be projected In the same way on to the 
hue, to the poinU b' c' and a' t' ; then B C and b V produced, 
win meet in n, and A C'and a' € produced, will meet in o. The 
two triangles A B C and a' b' t' are such, that the lines jobiag 
A to a', B to b', and C to d*, will, if produced, meet in a point, 
namely, the point on the base ABC which is the projection of D. 
Any two-triangles which fulfil this condition are the possible base 
and prpjecdon of the section of a pyrsmid; therefore the sides ef 
soch triangles, if produced in pairs, win meet 0^ they are not 
. paranel) In three points which Ue in one straight Une. 

A four-dimensional pyramid may be defined as a figure bounded 
by a polyhedron of any number of sides, and the same number ef 
pyramids whose bases are the sides of the polyhedron, and whose 
apices meet in a point not In the space of the base. 

If a four-dimensional pyramid on a tetrahednd base be cut by a 
qMce which passes through the four sides of the pyrunid in such 
away that the sides of the sectional figure be notparaUd to the 
•ides of the base ; then the sides of these two tetrahedrs, if produced 
in pabsi wUl meet m lines which an Ue hi one frfane^ namely, the 
pfame of intersection of the space of the base and the space of the 



\ 



i 



■< 



II 



If now the sectional tetrahedron be projected on to the base (by 
diawiog Unes firom each point of the section to the base at right 
angles to It), there win be two tetrahedm fitlfiUfog the condldon 
that the line joining the angles of the one to the angles of the 
other win, If produced, meet In a point, which point Is the projec- 
tion of the apex of the four-dimensional pyramid. 

Aay two tetrahedre which fulfil this condition, are the possible 
base and projection of a section of a four-dfanensional pyramid. 
Therefore^ in any two such tetrahedrst where the sides of the one 
aienotpanUeltothe sides of the other, the sides, if produced in 
pairs (one side of the one with one side of the other), wUl meet in 
tour straight lines which are aU In one plane. 



APPENDIX F. 

ExuciSBs ON Shapes op Threb DiiiBNSiONt. 

The names used are those given in Appendfat B. 

Find the shapes from the foUowing projections : 
I. Syce projections : RatIs, Caput, Castrum, PUigua. 
AItus projections : Merum, Ocidus, Fulmen, Pruinus. 
Moena projections : MUes, Ventus, Navis. 
3. Syce: Dies, Tuba, Lituus, Froos. 
Alms : Sagitta, Regnum, TeUos, Fulmen, Pruinus. 
Moena : Tibia, Tunica, Robur, Finis. 
3. Syce i Nemus, Sidus, Vertex, Nix, Cerva. 
Alvus! Lignum, Haedus, Vultus, Nemus, Humeros. 
Moena: Dexter, Princeps» £quus» Dux, Urbs, Puttis, Gens, 
Monstram, MUes. 
4 Syce : Amphora, Castrum, Myitns» Rota, Pafana, Mcta, Tmbs, 
Ratis. 
Ahms: Dexter, Princeps, Moena, Aes, Merum, Oculus,'Iitttts, 

Ciris, Fuhnen. 
Moena: Gens, Ventus, Navis, Fmis, Moastnim, Cursas. 
S* Syce : Castrum, Plagua, Nix, Vertex, Aper, Capul^ Cenra, 
Venator. 
Alvus : THumphus, Tellus, Caterva, Lignum, Haedus, PrufaittS, 
Fuhnen, CMb, Humerus, Vultus. 

Pharetra, Cursus, MUes, Equns, Dux, Navisi Mon- 
stium, Gens, Urbs, Dexter, 



206 



A N$w Era of Tlumghi. 



AMSWSR& 

The shapes are : 
I. UmbfBy Aether* Ver, Carina, Flos^ 
1. Pontns, Ciistos» Jaadttm, Pfatunit AnUoTi Agna. 

3. Focii% Onnis, Haedns, Tabema, Vultus, Hotds, Figwa, Alei, 

SiduSi Augur* 

4. Tcmpus, Campus, Fmisi Naris, Veotus, Pdagus, Notus, Cohort 

Aether, Carina, Res, Templum, Rex, Gens, Monstrum. 

5. Pertns, Anna, Sylva, Lucrum, Omus, Onus, Os, Fades, Chorus, 

Carina, Flos, Nox, Ales, Oamor, Res, Pugna, Ludut, 
Figure, Augur, Humerus. 

FURTHBE EXUCISES IN SHAPES OP THRBB DIMENSIONS. 

, ^ The Names used are those given in Appendix C ; and this set 
of exercises forms a preparation for their use in space of four 
AH are in the 37 Block (Uma to Syrma). 



Appendix. 



207 



I. Syce : Moles, Frenum, Plebs, Sypha 

AIvus : Ume, Frenum, Uncus, Spicula, Comes. 

Moena : Moles, Bidens, Tibicen, Comes, Saltus. 
X Syce: Uma, Moles, Plebs, Hama, Remus. 

Alvus : Uma, Frenum, Seaor, Ala, Mars. 

Moena : Uma, Moles, Saltus, Bidens, Tibicea. 

3. Syce : Moles, Plebs, Hama, Remus. 

Alms : Uma, Ostram, Comes, Spicule, Frenum, Sector. 
Moena : Moles, Saltus, Bidens, Tibicen. 

4. Syce : Frenum, Piebs, Sypho^ Moles, Hama. 
Alvus : Uma, Frenum, Uncus, Sector, SpicuhL 
Moena: Uma, Moks, Saltus, Scene, Vestts. 

5* Syce: Uma, Moles, Plebs, Hama, Remus, Sector. 

Alvus : Uma, Frenum, Sector, Uncus, Spicule, Comes, Mais. 

Moeitt : Uma, Moles, Saltus, Bidens, Tibicen, Comes. 
6b,Syce : Uma, Moles, Saltus, Sypho, Remus, Hama, Sector. 

Ahms : Cooms^ Ostram, Uncus, Spicub, Mars, Ala, Sector. 

MeeMi: Una, Moles, Saltus, Scene, Vestis, Tiblosn, Coowi, 
Ostfum* 
7. Syce : Sypho^ Sahns, Moles, Uma, Frenum, Seeter. 

Alvus : Um% Frenum, Uncus, Spicule, Mars. 

Moena: Saltus, Moles, Una, Ostram, Cenes. 
a. Syoe: Moles, Plebs, Heme, Sector. 

Ahmsi Ostram, Ficnom, Uncusb Spicule, Mai% Ala. 

Moena: Molesb Bidens^ Tibicen, Ostram. 



9k Syce i Moles, Saltus, Sypho, Plebs, Frenum, Sector. 

Alvus : Ostram, Comes, Spicule, Mars, Ala. 

Moena : Ostram, Comes, Tibicm, Bidens, Scene, Vesds. 
ta Syce : Uma, Moles, Saltus, Sypho, Remus, Sector, Frenum* 

Alvus : Uma, Qstram, Comes, Spicule, Mars, Ale, Sector. 

Moena : Uma, Ostram, Comes, Tibicen, Vestis, Scene, Saltus. 
II. Syce: Frenum, Plebs, Sypho, Hama. 

Alvus : Frenum, Sector, Ala, Mars, Spicule. 

Moena : Uma, Moles, Saltus, Bidens, Tibicen. 

Answers. 
The shapes era : 

I. Moles, Plebs, Sypho, Pallor, Mora. Tibicen, Spicuhu 
3. Uma, Moles, Plebs, Heme, Cortis, Merces, Remus. 

3. Moles, Bidens, Tibicen, Mora, Plebs, Heme, Remus. 

4. Frenum, Plebs, Sypbo^ Tergum, Oliva, Moles, Hama. 

$. Uma, Moles, Plebs, Hama, Remus, Pellor, Mora, Tibicen, 

Mars, Merces, Comes, Sectpr. 
6b Ostram, Comes, Tibicen, Vestis, Scene, Teigum, Olive, Tyro, 

Aer, Remus, Hama, Sector, Meroes, Mars, Ala. 
7* Sypho, Saltus, Moles, Uma, Frenum, Uncus, Spicule, Mais. 
& Plebs, Pallor, Mora, Bidens, Merces, Cortis, Ala. 
9. Bidens, Tibicen, Vestis, Scene, Oliva, Mora, Spicule, Mars, 

Ala. 
la Uma, Ostram, Comes, Spicule, Mars, Tibicen, Vestis, Oliva, 

Tyra, Aer, Remus, Sector, Ala, Saltus, Scene. 
II. Frenum, Pleb% Sypho, Heme, Cortis, Meraes, Mora. 



APPENDIX a 

Exercises on Shapes op Four Dimensions. 

The Names used era those given in Appendix C The firsi six 
exercises era in the <i Set, end the rest in the 356 Set 

I. Mak projection : Uma, Moles, Plebs, Pallor, Owtis, Merces. 
Lar projection : Uma, Moles, Plebe, Cun^ Penates, Nepos. 
Pluvium projection : Una, Moles, Vitte, Cudo^ Luctus, Trqje. 
Vesper p roj e ct ioii: Uma, Frenum, Cnttes^ Ocrea, Orcus, PMti% 
Aicus. 



208 



A Niw Era of Thought. 



% Mab: Uma, Frenttm, Uncus, Pftllori CMitt A«r. 

Lar : Unia, Fitiram, Cmtett Cmm, Ucus, Amiii AaginU. 
. Plnviiim: UfiiisTliyrNifl,Cardo^Cndo^Maltoiis,An^ 
Vctpcr : Uma, Frenum, Craldi OCTM, PUum, Portii. 

3. Mala : Comci, Tibicen, Mora, Pallor. 
Lar : Uma, Moles, Vitta, Cuia, Penates. 
PloYimnx Comcst Tibicen, Mica, Treia, Luctus. 
Vesper: Comes, Cortex, Praeda, Lannis, Orcus. 

4. Mala : Vestis, Otiva, Tyra 

Lar : Saltus, Sypho^ Remus, AnruSi Augusta. 
Pluvium : Vestis, Flagellum, Aries. 
Vesper : Comes, Spicule, Mars, Ara, Arcus. 

5. Mala: Mars, Meices, Tyro^ Aer,Tergum, Pallor, Plebs. 

Lar : Sector, Hama, Lacus, Nepos, Augusta, Vulcan, Penates. 
' • Pluyium : Comes, Tibicen, Mica, Troja, Aries, Anguis, Luctus, 
Securis. 
Vesper : Mars, Ara, Arcus, Postis, Orcus, Polus. 

6. Mala: PaUor, Mora, Oliva, Tyro, Meices, Mars, Spicule, 

Comes, Tibicen, Vestis. 
^ Lar: Plebs, Cura, Penates, Vulcan, Augusta, Nepos, Telum, 

Polus, Cervix, Securis, Vinculum. 
Pluvium : Bidens, Cudo, Luctus, Troja, Axis, Aries. 
Vesper: Uncu% Ocrea, Oicus, Lauras, Arcus, Axis. 

7. Mala : Hoepcs, Tribos, Fragor, Aer, Tyns Moia, Oliva. 

Lar : Hospes, Tectum, Rumor, Arvus, Augusta, Cera, Apis, 

La^ 
Ptavina: Acus, Torns^ Malleus, FlageUum, Thorax, Aries, 

Aestas, Ciqpella. 
' Vesper: Pardus, Rostnnn, Ardor, Pilum, Aia, Arcus, Aestus, 

Septum. 

8. Mala: Pallor, Tergum, Aer, Tjmn Cbitis, Syrma, Ursa, Fama, 

Naxos, Erisma. 
Lar: Pleb% Cura, Limen, Vukan, Augusta, Nepos, Cera, 

Papaver, Pignus, Messor. 
Pliivium; Bidens, CudObMaUenSpAnguisb Aries, Luctus, CapeUa, 

Rkeda, Rapina. 
Vesper: Uncus, Ocrea, Orcus, Postis, Arcus, Aestus, Cussis, 

Dolium, Alexis, 
a Mak: Faau, Conjux, Reus, Toras, Acus, Myrrius Sypbo^ 

Plebs, Pallor, Mora, Olhra, Alpis, Adas, Hircus. 
Lar: Mfaale^ Fortuna, Vka, Pax, Furor, Ira, Vulcan, Penatrs, 

Lapis^ Apis^ Cera, Pignus. 



• 1 



Appendix. 



209 



Pluvium : Torus, Plenum, Pax, Thorax, Dolus, Furor, Vincuhnn, 
Securis, Clavis, Gurges, Aestas, Capella, Corbis. 

Vesper : Uncus, Spicule, Mars, Ocrea, Cardo, Thyrsus, Cervix, 
Verbom, Orcus, Polus, Spes, Senex, Septum, Pdnrum, 
Cussis, Ddium. 

Answers. 
The shapes are : 
I. Uma, Moles, Plebs, Cura, Tessara, Lacerta, Qipeus, Ovis. 
a. Urna, Frenum, Crates, Ocrea, Tessara, Glaus, Colus, Tabubu 

3. Comes, Tibicen, Mica, Sacerdos^ Tigris, Lacerta. 

4. Vestis, Oliva, Tyres Pluma, Portio. 

5. Mars, Merces, Vomer, Ovis, Portio, Tabula, Testudo^ Lacerta, 

Penates. 

6. Pallor, Tessara, Lacerta, Tigris, Segmen, Portio^ Ovis, Arcus, 

Laurus, Axis, Troja, Aries. 

7. Hoepes, Tribus, Arista, Pellis, Cdus, Pluma, Portion Calathus, 

Turtur, Sepes. 

8. Pallor, Tessara, Domitor, Testudo, Tabula, Clipeus, Portion 

Calathus, Nux, Lectrum, Corymbus, Circaea, Cordax. 

9. Fama, Conjux, Reus, Fera, Thorax, Pax, Furor, Dolus, Scale, 

Ira^ Vulcan, Penates, Lapis, Palus, Sepes, Turtur, Diota, 
Drachma, Python. 



APPENDIX H. 
Sections or Cube and Tessaract. 

There are three kinds of sections of a cube. 

t. The sectional phme, which is in all cases supposed to be 
infinite^ can be Uken pandlel to two of the opposite faces of the 
cube ; that is, parallel to two of the lines meeting in Corvus, and 
cutting the third. 

3. The sectional plane can be taken parallel to one of the lines 
meeting in Corvus and cutting the other two^ or one or both of 
them produced. ' 

3. The sectional plane can be taken cutting all three lines, or 
any or all of them produced. 

Take the first case, and suppose the plane cuts Dos half-way. 
between Corvus and Cista. Since it does not cut Arctos or Cuspis, 
or either of them produced, it will cut Via, Iter, and Bolus at the 
middle point of each ; and the figure, de termined by the \amh 

P 



2IO 



A New Era of TlumghL 



sectkm of tlie Plane and Mala, U a square. If the length ol 
the c4ge of the cnbe be Uken as the vnit, this figure may be 

expressed thus: q . o . i showing that the Z and X lines 
from Corvus are not cut at all, and that the Y line is cut at half- 
a*unit from Conrns. 

Sections Uken ^ ^ Y ««> ^ o Y would also 
o.o.^ o«o«i 

be squares. 

Take the second case. 

Let the plane cut Cuspis and Dos, each at half-a-unit from Cor- 
YttS, and not cut Arctos or Arctos produced ; it will also cut through 
the middle points of Via and Callis. The figure produced, is a 
rectangle which has two sides of one unit, and the other two ate 
each the diagonal of a half-unit squared. 

If the plane cuts Cuspis and Dos, each at one unit from Corvus, 
and is parallel to Arctos, the figure will be a rectangle which has 
two sides of one unit in length ; and the other two the diagonal 
of one unit squared. 

If the plane passes through Mala, cutting Dos produced and 
Cuspis produced, each at one-and-a-half unit from Corvus, and Is 
paialld to Arctos, the figure will be a paraillelognun like the one 

obtained by the section o . | . X 

This set of figures will be expressed 



Z X Y 



Z 

o 



X 

I 



Y 

I 



Z X Y 



It will be seen that these sections are parallel to each other; 

and that hi each figure Cuspis and Dos are cut at equal distances 

from CorvnsL 

We may express the whole set thus':— 

Z X Y 

O.I.I 

it betof understood that where Roman figures areuscd, the numbers 

do not refer to the length of unit cutoff any given line from Conros, 

Z X Y 
but to the proportion between the lengths. Thus q . i .• h 

means that Aictos is not cut at all, and that Cuspis and Dos are 
art, Dos being cut twice as iar from Conrus as is CuspiSi 

These figures will also be rectangles. 

Take the thirds 



Appendix. 



211 



Suppose Arctos^ Cuspis, and Dos are each cut half-way. This 
figure is an equilatetml triangle, whose sides are the diagonal of 



a half-unit squared The figure 



X 

I 



X 



\ Is also an equi- 
I is an equilateral 



lateral triangle, and the figure ^x 

hexagon. 

It is easy for us to see what these shapes are, and also, 

Z X Y 
what the figures of any other set would be, as j n • II 



but we must learn them as a two-dimensional 



X Y 

"'I . II . Ill 

being would, so that we may see how to learn the three-dimensional 

sections of a tessaract 

It is evident that the resulting figures are the same whether we 
fix the cube, and then turn the sectional plane to the required 
position, or whether we ^ the sectional plane, and then turn the 
cube. Thus, in the first case we might have fixed the plane, and 
then so pkured the cube that one plane side coincided with the 
sectional plane, and then have drawn the cube half-way through, in 
a direction at right angles to the plane, when we should have seen 

the square first mentioned. In the second case ^^ ^ y^ 

we might have put the cube with Arctos coinciding with the plane 
and with Cuspis and Dos equally inclined to it, and then have 
drawn the cube through the plane at right angles to it until the 
lines(Cuspis and Dos)were cut at the required distances from Corvus. 
In the third case we might have put the cube with only Corvus 
coinciding with the plane and with Cuspis, Dos, and Arctos equally 

inclined to it (for any of the shapes in the set | ^ ^ ^\ 

* and then have drawn it through as before. The resulting figiures 
are exactly the same as those we got before ; but this way is the 
best to use, as it would probably be easier for a two-dimensional 
bemg to think of a cube passing through his space than to 
imagine his whole space turned round, with. regard to the cube. 

We have already seen (p. 117) how a two-dimensional being 
would observe the sections of a cube when it is put with one pbne 
side coinciding with his space, and is then drawn partly through. 

Now, suppose the cube put with the line Arctos coinciding with • 
h» qiace, and the lines Cuspis and Dos equally inclined to it At 
firM he woukl only see Arctos. If the cube were moved until 
I><M and Cuspis were each cut half-way, Arctos still behig paiallel 



313 



A New Era of Thotight. 



to the pliuM^Aictos would disappear at once; and to find out what 
he would see he would have to take the square sections of the cube, 
and find on each of them what lines are given by the new set of 
sections. Thus he would take Moena itself^ which may be re- 
. garded as the first section of the square set. One point of the 
figure would be the middle point of Cuspis, and since the sectional 
phuie is parallel to Aictos, the line of intersection of Moena with 
the sectional phuie will be parallel to Arctos. The required line 
then cuts Cuspis half-way, and is parallel to Arctos^ therefore it 
cats CalUs half-way. 

Next he would take the square section half-way between Moena 
and Mufex. He knows that the line Alvos of this section is 
parallel to Arctosi and that the point Doe at one of its ends is 
half-way between Corvus and Cista, so that this line itself is the 

* one he wants (because the sectional plane cuts Doe half-way 
between Corvus and Cistay and is parallel to Arctos). In Fig. 3i 
the two lines thus found are shown, a b is the line in Moena, 
and c d the line in the section. He must now find out how far 
apart they are. He knows that from the middle point of Cuspis 
to Corvus is hal-fa-unitt and from the middle point of Dos to 

* Corvus is half-a-unit, and Cuspis and Dos are at right angles to 
each other ; therefore from the middle point of Cuspis to the 
middle point of Dos is the diagonal of a square whose sides are 
half-a-unit in length. This diagonal may be written d (()*. He 
would also see that from the middle point of Callis to the middle 
point of Via is the same length ; therefore the figure is' a parallelo- 
gram, having two of its sides, each one unit in length, and the 
odier two each d(i]^. 

He ooukl also see that the angles are right, because the lines 
a c and b d are made up of the X and Y directions, and the 
odier twq^ a b and c d, are purely Z, and since they have no tendency 
In comoMNi, they are at right angles to each other. 

If he wanted the 4gure made ^ o • i4 • i4 '^ ^"^"^ ^ ^ 
little moie difficult He would have to uke Moena, a sectkm half- 
wiqr be t wee n Moena and Murex^ Murex and another square which 
he would have to regard as an imagimary section half-a-unit 
lurther Y than Murex (Fig. 33). He might now draw a ground 
plan of the sections ; that is, he woukl draw Syce, and produce 
Cuspis and Dos half-a-unit beyond Nugas and Cista. He wookl 
see that Cadns and Bolus would he cut half*wayi so that in the 









i 



M« 


\ 1 

fMT 


i 


' 



Isiyiasgri 

StiVM \ 



r«.?2. 



ITV/iflr^ tit. 









CtftlHt 








*•••*• 



I tmtgMf 



Fi5.24. 



ir>0rA.m|> 



Appendix. 



21$ 



half-way section he would have the point a (Fig. 33), am) in Murex 
the point c In the imaginary section he would have g ; but this 
he might ditregaid, since the cube goes no further than Murex. 
From the points c and a there would be lines going Z, so that Iter 
and Semita would be cut half-way. 

He could find out how faa the two lines a b and c d (Fig. aa) 
are apart by referring d and b to Lama, and a and c to Cms. 

In taking the third order of sections^ a similar method may be 
followed. 

Suppose the sectional ptone to cut Cuspis» Dos, and Aicto% 
each at one unit from Corvus. He would fint take Moena, and 
as the sectional phme passes through Hex and NugK^ the line 00 
Moena would be the diagonal passing through these two points 
Then he woukl take Mttrex, and he woukl see that as the plane 
cuu Dos at one unit from Corvus, all he woukl have is the point 
Cista. So the whole figure is the Ilex to,Nugae diagonal, and the 
point Cbta. 

Now Cista and Hex are each one 'inch from Gntvus, and 
measured along lines at right angles to each other ; therefore^ they 
are d (i)' from each either. By referring Ni^;» and Qi^ to 
Corvus he would find that they are also d {if apart ; theiefoie the 
figure is an equilateral triangle^ whose sides are each d {if. 

Suppose the sectional plane to pass through Mah^ cutting Cuspis, 
Dos, and Arctos each at i^ unit from Corvus. To find the figure, 
the plane-being would have to uke Moena, a section half-way 
between Moena and Murex, Murex, and an imaginary section half-a* 
unit beyond Murex (Fig. 34). He would produce Aictos and Cuspis 
to poinu half-a-unit from Ilex and Nugae, and by joining these 
points, he would seo that the line passes through the middle points 
of Callis and Far (a, b. Fig. 24). In the Ust square, the imaginary 
section, there woukl be the point m ; for this is i^ unit from 
Corvus measured akmg Dos produced. There would also be lines 
in the other two squares, the section and Murex, and to find these 
he would have to make many observations. He found the points 
a and b (Fig 34) by drawing a line from r to s, r and s being each 
i\ unit from Corvus, and simply seeing that it cut Callis and Far 
at the middle point of iach. He might now imagine a cube Maki 
turned about Arctos, sa that Alvus came into his pUne ; he might 
then produce Arctos and Dos until they were each i| unit long, 
and join their extremities, when he woukl see that Via and Budna 
Me etch cut half-way. Again, by tumtog Syce into his planc^ and 



214 



A New Era of Thought. 



prododng Dot and Cuspit to poinU i\ unit from Corvus and 
' joiaiag the pointSi he would tee that Bolus and Cadut are cut half- 
nucf. He has now detennined six points on Mala, through which 
the pbne passes, and hy referring them in pairs to Ilex, Olus, 
Cista, Cms, Nugae, Son, he would find that each was d (()* from the 
next ; so he would know that the figure is an equilateral hexagon. 
The angles he would not have got in this observation, and they 
might be a serious difficulty to him. It should be observed that 
a simitor difliculty does not come to us in our observation of the 
sections of a tessaract : for, if the angles of each side of a solid 
4gure are determined, the solid angles are also determined 

There is another, and in smne respects a better, way by which 

he might have found the sides of this figure. If he had noticed 

. his plane-space rouch, he would have found out that, if a line be 

' drawn to cut two other lines which meet, the ratio of the paru of 

the two lines cut 00" by the first line, on the side of the angles is 

the same for those lines, and any other two that are parallel to 

them. Thus, if a b and a c (Fig 35) meet, making an angle at a, 

and b c crosses them, and also crosses a' b' and a' c*, these last 

: two being paiaUei to a b and a c, then a b : a c : : a' b' : a' c'. 

If the plane-being knew this, he would rightly assume that if 
three lines meet, making a solid angle, and a plane passes through 
them, the ratio of the parts betwee n the fdane and the angle is the 
same for those three lines, and for any other three parallel to them. 
In the case we are dealing with he knows that firom Ilex to the 
point on Arctos produced where the plane cuts, it Is half-apunit ; 
and as the Z, X| uid Y lines are cut equally from Corvus, he would 
conclude that the X and Y lines are cut Uie same distance from 
Ilex as the Z line^ that is half-a-unit He knows that the X line 
Is cut at 1^ units finom Corvus ; that is, half-a-uait from Nugae : 
so he would conclude that the Z and Y lines are cut half*a*unit 
from Nn^gae. He would also see that the Z and X lines from Cisu 
are cut at half-apuoit He has now six poInU on the cube, the 
middle poinU of Callis, Via, Budna, Cadus, Bolus, and Far. 
Now, looking at his square sections, he would see on Moena a 
fine going from middle of Far to middle'of Callis, that is, a line 
d {^f kmg. On the section he wouki see a line from middle of 
Via to middle of Bokis d (i)> long, and on Murex he wouU seea 
Bae from nuddle of Cadus to nuddle of Budna, d (i]^long. Of 
these difee lines a b^cd,ei; (Fig. S4)'^a band e fare sides, and 
cdisasectioaof the required figure. He can find the distances 




(r#>iflr^M4. 



Appendix. 



215 



between a and c by reference to Ilex, between b and d by refer- 
ence to Nugse, between c and e by reference to Olus, and between 
d and f by reference to Crus ; and he will find that these distances 
are each d (^)* . 

Thus, he would know that the figure is an equilateral hexagon 
with its sides d (^)' long, of which two of the opposite points (c and 
d) are d (1)' apart, and the only figure fulfilling all these conditions 
is an equilateral and equiangular hexagon. 

Enough has been said aboi^ sections of a cube, to show how a 

plane-being would find the shapes in any set as in , \i .^ 
Z X Y I . II . II 

®' I . I . II 

He would always have to bear in mind that the ratio of the 
lengths of the Z, X, and Y lines is the same from Corvus to the 
sectional plane as from any other point to the sectional plane. 
Thus, if he were taking a section where the plane cuts Arctos and 
Cuspis at one unit from Corvus and Dos at one-and-a-hali^ that 
is where the ratio of Z and of X to Y is as two to three, he would 
• see that Dos itself is not cut at all ; but from CisU to the point 
on Dos produced is half-a-unit ; therefore from Cista, the Z and X 
lines will be cut at f of ^ unit from Cista. 

It is impossible in writing to show how to make the various 
sections of a tessaract ; and even if it were not so^ it would be 
unadvisable ; for the value of doing it is not in seeing the shapes 
themselves, so much as in the concentration of the mind on the 
tessaract involved in the process of finding them out. 

Any one who wishes to make them should go carefully over the 
sections of a cube^ not looking at them as be himself can see them, 
or determining them as he, with his three-dimensional conceptions, 
can ; but he must limit his imagination to two dimensions, and 
work through the problems which a plane-being would h^ve to 
work through, although to his higher mind they may be self* 
evident Thus a three-dimensional being can see at a glance^ 
that if a sectional plane passes through a cube at one unit each 
way from Corvus, Uie resulting figure is an cquilatend triangle. 

If he wished to prove it, he woukl show that the diree bounding 
lines are the diagonals of equal squares. This is all a two- 
dimensional being would have to do ; but it is not so evident to 
him that two of the lines are the diagonals of squares. 

Moreover, when the figure is drawn, we can look at fc from a 
point outside the plane of the figure^ and can thus tee it all at 



3l6 



A New Era of Thought. 



frnae ; bat he who has to look at it from a point in the plane can 
imly tee an edge at a time, or he might tee two edges in perspec- 
Itve together. 

Then there are certain suppositions he has to make. For 
instance^ he knows that two points determine a line, and lie 
assumes that three points determine a planer although he cannot 
conceive any other plane than the one in which he exists. We 
atsone that fonr points determine a solid space. Or rather, we 
say that 9/ this supposition, together with certain others of a like 
nature, are true, we can find all the sections of a tessaract, and of 
other four-dimensional figures by an infinite solid. 

When any diflkulty arises in taking the sections of a tessaract, 
the surest way of' overcoming it is to suppose a simiUr difficulty 
occurring to a two-dimensional being in taking the sections of a 
cube, and, step by step, to follow the solution he might obtain, and 
then to apply the same or similar principles to the case in point 

A few figures are given, which, if cut out and foMed along the 
fines, will show some of the sections of a tessaract But the reader 
» earnestly b^ged not to be content with looking at the shapes 
9nly. That will teach him nothing about a tessaract, or foiir^ 
dimensional space, and will only tend to produce in his mind a 
feeling that ^the fourth dimension" is an unknown and unthink- 
able region, in which any shapes may be right, as given sections 
of its figures, and of which any statement may be true. While, in 
feet, if it is the case that the hiws of spaces of two and three 
dimensions may, with truth, be carried on into space of four 
dimensions ; then the little our solidity (l>he the flatness of a 
plane-beiog) will allow us to learn of these shapes and relations, 
b no more a matter of doubt to us than what we learn of two- and 
three-dimenskNial shapes and relations. 

There are given also sections of an ocu-tessaract, and of a 
tetm-tessaract, the equivalents in four-space of an octahedron and 
tetrahedron. 

A tetrahedron may be regarded as a cube with every alternate 
eomer cut olll Thus, if Mala have the comer towards Corvus cut 
off as fer as the points Ilex, NugK, Cista, and the comer towards 
Sors cat off as fer as Ilex, NugK, Lama, and the comer towards 
Cras cut off as fer as Lama, Nugae, Cista, and the comer towards 
Ofes cut off as fer as Ilex, Lama, Cista, what is left of the cube is 
^ whose angles are at the points Ilex, Nugae, Cista, 
In a sfaniiar manner, if every alternate comer of a tessaract 




(r#>MiAtilk^M.i.*vl« 



A 



>v 




I 







Appendix. 



7X1 



be cvt ofl; the figure that is left is a tetra-teMenct, wliicfa is a 
figure bounded by sixteen regular tetrahedrons. 

The octa^tessaract is got by cutting off every comer of the 
tessaract If every comer of a cube is cut off, the figure left is 
an octa-hedron, whose angles are at the middle points of the sides. 
The angles of the octa-tessaract are at the middle points of its plane 
sides. A careful study of a tetra-hedron and an octa-hedron as 
they are cut out of a cube will be the best preparation for the study 
of these four-dimensional figures. It will be seen that there is 
much to ieam of them, as— How many planes and lines there are 
in each, How many solid sides there ait round a point in each. 



A DBSCRiPTioif or Figures a6 to 41. 



W 



9 V V _ f 36 is a section taken i . 1 • 1 . r 

, , , 7 {27 li . ij . ij . x\ 

*•'•'•* (a8 a . a . a . a 



Z 
II 



X 

II 



Y 

II 



W 

I 



^ 39 is a section taken i 
\ 30 ••• ••• ••• 1^ 

(31 a 

32 «•• •%% ••• %\ 



X 


Y 


W 


1 . 


1 . 


i 


'i. 


•i . 


1 


a . 


3 . 


I 


a*. 


a* . 


•i 



The above are sections of a tessaract Figures 33 to 35 are of 
a tetra-tessaract The tetra-tessaract is supposed to be imbedded 
in a tessaract, and the sections are taken through it, cutting the Z 
X and Y lines equally, and corresponding to the figures given of 
the sectkms of the tessaract 

Figures 36, 37, and 38 are similar sections of an octa-tessaract 
Figures 39^ 4<^ •nd 41 Aie the following sections of a tessaract. 

Z X Y W 

section taken o • ^ . ^ . ^ 

o • I . t • 1 

•• o . i| • 1^ • i| 



Z 
O 



X Y ^l^iiftiectio 
I . I . I jit *^ ••• 



It is clear that there are four orders of sections of every foui^ 
dimensional figure ; namely, those beginning with a solid, those 
beginning with a plane, those beginning with a line^ and those 
banning with a point There should be little dilBculty in finding 
them, if the sections of a cube with a tetra-hedran, or an octa* 
hedron enckMed in it, are carefully examined. 

Q 



->w/-.-:- 



PARTIL APPENDIX K. 



Model i. MALA. 




CoLOVts I Mala. Lioht-bofp. 

/VMr: Connis, GoM. Ni^gK, Fswn. Cnu^ Tcnm-«dtta. Cbta, Bnfll 
Ilex, Ucht^UM. 8«s» D«U-piifple. Lum« DMp>blMi Olnt, 
Red. 

Ubm: CMtpis, Onunge. Boku, CrimflOB. Oidei» Green-gref* Dot, Blee. 
Aicto% Brown. Far, Freach-gicj. Dapi, Daric-tlate. Bttcfau, 
Green. Geliis, ReddUi. Iter, Bi^hl-blne. Searita, Leeden. 
Via, Deep^ellow. 

Smfautt Moena, Dark-blae. Proes, Bine-green. Maveit Vn^'fiDiam* 
Alnup VcmiUon. Mel, Whte Sycc,r 



PART II. APPENDIX K. 



MoDita. MAROa 




•r" 



8M«« 



Lvt* 



COLOVBS t MAIOOk SAOB-OItSM. 

Firims: Splfa, Sihner. Aadllft, IVtiqaoiie. Ifsgfl, BortlMii, Faimk, 
Btae-tint Fdfab QwilMr-graeii. FiiMr^ P^i€Ock*Uiit. T«Juf« 
Oimge^veroiilios* SoUft» Pwplt» 

Ihutt Lnct, Leaf-green. Mappt, Dott-grecn. Me«mnit Dark*porple, 
Opex, Plirple*browii. Pagw» D«U*blttt. OiMgw* Dftffc-plnk. 
Vena, Pale*plak. Lixa,LidlffOk Tlioltttt Bfowa-gmin. Cak)r» 
Daik-green. Ufor^ Mt-jptUmr. LtiM» Dirk 

Stufmcm: SUex, Bornt-itonna. Sal, Ytlloir<MlMa. Portlea» Dim. Cn»t . 
Indlaa-nd. Lai«% Ught-gnf* Lappa, IW^ (pBW. 



PART IL APPENDIX K. 



MoMLj. LAR. 




CoLovmss LAit BucK-ua 

Ptmtt: Spira» Silver. AncOla, TWrqaoba Misil» Ettdwik P)mmx» 
Blne-tbit Cofvi% GokL Nvga^ Fawa. Grai» TcfiKotUL 
CbUfBttAl 

LUtet: Lwftt Letf-greea. Mappa, IHiU*peeii« McnNn» I>ttk*parpl«. 
Opes, Perple-lifoinb Op% Stooe. Linrai* Sao fc fc dfinr. 
Magenta. Li% Mght-gnen. GaipK OiMge. BokM^ Ciioi 
. toB. Cadea» Giceo-gRej. Dos, Blaa. 

Sm^km: Lotw, Ame. OUa, Rote. Lorica, Sca-bl«% Lm, Bdglit* 
Mvwn* Sjce, B la cfc i Lappa^ Br^gw^gien, 



PARTIL APPENDIX K. 



ICoDit4. VBLUIL 




TalM 



Tk«lv« 



C6u>0BS I Vblvk, Oiocolatl 

Mnis: Fdisi Qmkcr-green. Paster, Peacock-blae. Tahis, Orange^ 
TemiiUoB. Solk, Purple. Hex, Llgfac-blsa. 8<nb» DuU* 
puiplt. Laom, Dmp-Um. Olasi Red. 

Lhmi nblnfl, Brown-green. Calor, Dtrk-green. Livor, Pale-yelloir. 
LenMt Daik. LnoU, Rich-red. Pktor, Green-bine. libeim* 
Sen-ginen. Om, Enendd. Cidlls, Reddfah. Iter. Bf(g)it- 
blnei SemiUf Leaden. Via, DeepTettow. 

SmfKmt LimlM% Odue. Meatui, Deep-brown. Mango, Deep-gvML 
Qoela, X4g9itfnd. Md, WUu. Lame, Xig|it-gf^. 



PART II. APPENDIX K. 



Model 5. VESPER. 



V*^^* 




Ct«cU 



C^" 






CWM««n« 






*<' 



■*T5F 



i 



^•^ 



COU>ORS S ViSPUIt PALB-GKBBir. 

Brimhi Spin, SUfcr. Comi% Gold. CisC% Bsfl. Piauc, BlM>liat< 
Fe»% Qnaker-grecB. Iks» Light-Um. Oks, R«d. SoUa, 
Purple. 

ZImt; Opt* Stoned Dot, Bine. Lis, Ught-creea. Opei, Parple-browa. 
Pigui, Dnll-blve. Aretoe, Brown. BiiciM» Green. Uxe, 
Indigo. Lncte, Ridi-nd. Vie. Deep-TcUow. OiMpEmenld. 
LcnetyDeilc. 

Smfitm: Fticinit YeUow. Alms, Vennttion. Cemoene, Deep>oiaeoB. 
Cras, Indiea-red. Croelev Lig^-red. Lne, Mglrt-bfOim. 



PART II. APPENDIX K. 



MoraLtfw IDUS. 




CouMiRS I Iotft» Oak* 

Mmit: AacHk, Tiifq«oifl& Nvga^ Fawiv Cnifi Tcnm-cotta. M«gft 
Earthen. Pa«er» PeMock-Unei 8oii| D«U*p«ipliii Lmm* 
Deep-bhie. Talii% OfBOfe-TemQioik 

litm: Lfanus, Smoke. Boln% Crimioii. Ofli»» BUgenla. MappAt 
DuU-graen. Onagert Dark-piak. Far« Fnnch-gicjr. I>h»» 
Daik-alate. Vena, Pale-pftak. PMor« GfMA*blM. Iur«Bf%lil« 
bine. libera. Sea-green. Cakv. Daik-gieen. 

Smftm: Pi^tnai, Yellow-green. Pioeii, Bl e e g ree n . Ofea» I>aik«gnf* 
Salt YeDow-odiie. Meatn^ Deep-bmro. Olli» Roae. 



PART II. APPENDIX K. 



f*«* 



MOOBLT. PLWIUM. 

% CtttH ^ 








JUi 



^ 



<^•'••^ 



Ctf '' L^u* 



^ 



Liic* 



GOUNItf I PLUV10IC» DAXK-nONB. 

/Vter; Spiia, Sil?er. AncUte, Tniqttobe. Vwgm, Fawa. ComM^ 
Gold. Fella, Qnakcr-srccn. Fluier, Pcaoock-blm. Son» 
Dnll-pvrple. Ilex, Light-bloe. 

Uma ; LMi, Leaf-green. Lima% Snoke. Ottpis, Oniige. Opi» 
Stone. Pegu, Dall-Uee. Onager, Dark-pink* Far, Fiench* 
grey. Aretoa, Brown* Tkoloi, Brown-green. Pator, Green* 
Unei OOlii, Reddidk Lacta, Rk^icd. 

Smibm: SUex, Bamt-Sienna. Pactnn^ Yell e w gr e e n. Ifoena, Daifc- 
blM. P)i«inn, Yeiloir. Lfanbne, Odm. LoCnib Anre. 



.•»* 



PART 11. APPENDIX K. 



Model t, TELA. 




COLOOIS I TtLAt SALMON 

AMrf Puwx, Bne-tint Unffl^ Earthen. CnH» Tcm-eotte. Oita, 
Bnit SotSa, Piirpl& Taivs» Omnge-Tcmillioii. Lam, 
Deep-Uue. 01as» Red. 

JUim: ICeneafa, Dark-purple. OSex^ Bfagenta. Cadui, Gtcen-giey. 
Lis» Light-green. Uxa, Indigo. Vena, Pale-pink. Dapf, 
Daik-slate. Bndna, Green. Livor, Pale-jreUoir. Libera, 
Sea-green. Scnita, Leaden. Ona, Bmendd. 

Ams^Smv: Poftica, Dnn. Orca, Dark-grey. Meicx, Ligfal-fdk>w. 
Camoena, Deep^rfanaon. Mai^go^ Deep-gieen. Lofka, Sea^ 



PART IL APPENDIX K. 



MOMt 9. SECTION BETWEEN MALA AND MARGO. 



U^ 



S H««« 



I ^''• 



H^Phiwm 



i ^•••••Rffiffk^K »«« 



/• 









J 




<^" 



L«»»« 



GoLOVM s Intbuor <m TKtsAiACTt Wood. 

iWMrCriMr)! Opt» Slone. Uim% Snolie. Ofla. MagMiiL Li% 

L^t-grecD. LMta. Rich-rad, FMor» deoi-blMb Ubcn, 

SeH'gnuL OrM, EnenUd. 
£ter (A(y/&Mr) t LoCsi» Aare. Olla, Rom. Loricft* ScA-blaai Lna 

Bright-bfOfni. Paging Yellow. PMlany YtOow^giwo. 

Oict, Dwk-gny. QiMoeiMi, Doep-crioMOB. IMmm, Ochm, 

McatMb Deep*biowB. Kugo^ Dcep-greok Ciocte» I4g|hl 

nd. 
AngiiMflr (MUr) 1 Ftorhuit Duk^Hoiie. Id«% Oak. Tda, 

Voip<f> Palo gffoaa. Vilm, CiMOolalo. Lar« BMok«i«d. 



PART IL APPkNDIX K. 



^ 



MOML to. SEOnON BETWEEN LAR AND VELUM. 



^^i 



Mifrfii 



^^ 



MaU 



I T^ 



^f Pllvvfvm 









^ 



/ 



f 



Jtm9g9^ 



•Sir 




^ 



r 



I 



i 



CoMNFis t IirmtiOE oe Tmsaract, Wood. 

iWi«r(£tfMr)t F^«w,DBD.blM. Outer. Dwrk-piak. Vent, Ptola-ptak. 

LIm, ladigA Arctot, Btowb. Far, Fnacli-gnjr. pi^pii 

DtflctlAtc. Biidna» Greta. 
ZAMy(JbsfCMtf) s SUest Bunt-iiauuu Sal* Ydlow^oduc Portiea, 

Dan. Cms. IndiaBied. FHflM, YcUow. Pkctem, YcUow^ 

(reen. Orctf Dark«grcy. CeiaofiHi Dee p cr imt oD. If ocne^ 

Derk-bltte. Proeib BlaegrwiL Mani» Ii|^«jfiIloir. AIyih^ 

VermUion. 
Smfmi^Uait)i PkftaM, Daik-eloMb Idw. Oidc. Tdt, Sdnoa. 

VftpiTp Palt-giMB. Mida, LVu-bUC Maiga^ Sag e g wau 



PART It. APPENDIX K. 



Mtao. It. SlCnON BXTWBBN VESPBR AND IDUt;. 



<v 




jisrtgfL 



jf 



i 



ufff 



Pi !•«)•» 



/ 



L: 



/ 



>-: 









L«lv« 



COLOVIS I IMTBIIOK OR TtStAlACT. WOOO. 

AMr(ZAMr) t Loctt Leaf-green. Qupis, Orange. Cada% Gieen-giejr. 
Menswa, Daik-pnrple. ThoIa% Brown-green. Calllii Red* 
diih. Semlta* Leaden. LlYor» Palc-feUoir. 

Lkm {fimfiua) t Lotas, Anie. Syoe, Black. Loricat Sea-bhie. Lap* 
pa. Bright-green. SQex, Bomt-elcnniL Mocna, Dark-bine. 
Morex* Li|^-jdIo«r. Ptortiea, Dun. Llmlia% Ockm> lCd» 
White. Mango, Deep-green. Lareiy Ligfat«grcy; 

Smfam{fiMb)i Plnfiam, Daik-itone. Mak. Light-bnft Tck, Sal- 
nMtt. Karfo^ Sage green, Veknn, Chooolale» Lar, Brick* 



r\' 



PART 11. APPENDIX K. 
II. SECTION BBTWEEN PLUVIUK AND TELA. 



vJ 




COLOOBSt iNTBRtOR OR TittARACTi WOOD. 

AnMSr (ZAmv) t Opex, Pniple-bvowR. MapjM, Dull-fifMn. Boltti» 
Crimon. Dot, Blue. Lcma, Dark. Calor, Dark-greea. 
Iter, Briglit-bliie, Via, Deep-yellow. 

Zitm {Su^fiua) s Lappa, Bright-green. OUa,.RoM. Syce, Black. Laa, 
Bright-brown. Cmi, Indian-red. Sal, Ydk>w-ocluna. Protib 
Blae-green. Alvns, VeimilkMi. Larei, Ligfat-gny. McatMb 
Deep-brown* Mel, WUte. Oroeta, Ligbt-nd. 

S^frfiM'(S$Udi)i llaiga, Sagfgraen. Idai^ Oak. Mah, L^ht-baA 
V«pcr9 Pale green. Velaa^ Oioooiate. Lu, Brfdi-iad. 



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